Notes on Ballistics

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THEFOLLOW INGNOTESHAVEB'EEN'RECEIVED FROM AN

:OFFICER W"ITHmE AMERICAN EXPEDITIONARY FORCE; .IN FRANCE, AND ARE OFFERED AS SUGGESTIONS FOR

EQUIPMENT BEFORE GOING ABROAD.

Study French as much as possible.Settle all business before you leave as it is a long time b~-

tween mails here. It is hard to get to Paris or London, even Ifthere is nothing to do. , .. ....

Make arrangements for a checking account with some bank;The Guaranty Trust Company or Farmers' Loan and Trust Com-pany, New. York. City, suggested.

Get Western Union Code .. Register code addre~s with theWestern Union Company and the Naval Censor.

Payma~ters at home demand. date of sailing before they willpay foreign service increase on home vouchers. This is impossibleto give until months later, so keep that pittance here for pocketmoney.

At present, One Dollar equals f5.70 or 4 Shillings, 2 Pence.

Bring American money for 2 months' supply.

Have supply of American stamps.

Wardrobe trunks and wardrope steamer trunks get by, andfor Artillery Staff Officers are very handy.

Mark all equipment, all over.Officers carry two foot lockers, one bedding roll and a small

hand bag-the first always comes into camp late and the toilet~rticles, etc., are convenient and can be carried by buglers.

Keep your gold medal cot in your own bedding roll.

Caps are good on transports, but are not worn in France.

Supply yourself with the following articles: Flint-wheel andtinder cigar lighter; knee laced water proof boots; canned solidifiedalcohol; American tobacco for two months; gasolenevapor lantern(Montgomery & Ward); a good fountain pen; s~me chocolate; con-centrated soup tablets, or small cans of soup for an emergency;fleece lined gray leather gloves are good.

Sam Brown belts are handled by U. S. Ordnance Department,Paris, at $6.00 to $7.00.

Trench coats in England and France cost $35.00 for the best.

Whip cord breeches with doe skin knees cost $6.00 a pair in. E ngland or France, and are almost weatherproof.



Have all men fully .equippedbefore you leave.Have extra' hats for those blown overboard 'and also extra

mess outfits.Box up ph;;tols until you reach finai destination.

Make the men allot and deposit money.Everyone is paid in French money at f5.70 for one Dollar.

Have stationery and blank forms of all kinds.

Take plenti ful supply of shoe and leggin laces;Have each man load up with handkerchiefs, tobacco, face and

laundry soap, and one novel each. If he hasn't money, use Com-

pany fund.Have sweaters, mits, woolen helmets, etc., for officers and

men. The Americans in France have same rations as at horne.Keep two to five days' canned rations with organization at all

times, even if" carried in spare barrack bags.Bring supply of coffee-there is plenty of tea in England but

they do not like coffee as we do.. Have about 20 wash basins for men to wash face and hands.

Men must bathe in: creek.Have collapsible table and camp stools for office.

'



NOTES ON BALLISTICS

DIRECT FIRE

By Captain George A. Wildrick, C. A. C.

HIGH ANGLE FIREBy Lieut.-Colonel Alston Hamilton, C.A. C.

Corrected reprints from the Journal of the United States Artillery forJanuary-February, 1915, and September-October, 1913

Price, $0.50 postpaid

Third Edition

Journal U. S. ArtilleryFort Monroe, Virginia

1917



COPYRIGHT

BYJOURNAL U. S. ARTILLERY



Direct Fire

(Reprint from JOURNAL U. S. ARTILLERYforJanuary-February, 1915.)

By 1ST LIEUTENANT GEORGE A. 'WILDRICK, COASTARTILLERY CORPS

For two or three years past there has seemed to be aneed for an article on direct fire, both because of new methodsof application of ballistic data and because of works hithertoin use being out of print. This article treats only of the prob-lems most likely to be encountered at the battery: for athorough study of the subject, there are suggested the follow-

ing standard works, from which the present writer has freelyborrowed:

Artillery Circulars N, Exterior Ballistics, and M, BallisticTables, by Co!. James M. Ingalls, U. S. A., retired.

Ballistics~ Part I, by Major Alston Hamilton, CoastArtillery Corps.

Notes on. Ballistics, by Major Alston Hamilton,. CoastArtillery Corps, JOURNAL OF THE U. S. ARTILLERY, Nov.-

Dec., 1909.Manual for Coast Artillery, Part I, Gunnery, D. R. C. A.,

1905.For high angle fire, see Notes on Ballistics, by Major

Alston Hamilton, C. A. C., published in the JOURNAL OF THEU. S. ARTILLERY,Sept.-Oct., 1913.

The writer desires to acknowledge his indebtedness toMajor J. M. Williams, C. A. C., Major George A. Nugent,

C. A. C., and Capt. A. L. Rhoades, C. A. C., for encourage-ment and advice.

(1)



Seconds.Miles per hour.l\liles per hour. (In some

computations, as' in de-duction of wind. factor,wind is expressed in feetper second as

W = ~6\Vx.)

pounds.

Pounds; m ~ .gFeet per sec. ; 32.16 is assumed

as the standard.Feet per second per second.Pounds.Seconds.

Degrees, minutes, and tenths

of a minute.

Degrees, minutes, and tenthsof a mInute,

Degrees, minutes, and tenthsof a minute.Degrees, minutes~ and tenths

of a minute.Degrees, minutes, and tenths

of a minute.Thus, if we know cp for a

2250-f.s. gun and a range of

7000 yds., we denote bylp ' the angle of .departurewhen the M.V. IS reducedto, say, 2200 f.s.

Degrees, minutes, and tenthsof a minute.

Expressed inDegrees, minutes, and tenths

of a minute.Feet per second.Feet per second.Feet per second.

DIRECT FIRE

Inclination of trajecto-ry to horizontal atany point.

Angle of fall. (Some-times expressed interms of slope, as 1on n.)

Weight.

Mass.Acceleration due to

gravity.Retardation.Resistance.Elapsed time after gun

is fired.Time of flight.""ind velocity.Range component of

wind velocity; plusfor rear and minusfor head wind.

Quadrant angle of ele-vation.

Denoting a non tabular

value.

Muzzle velocityRemaining velocity.Velocity at point of

fall.Range table angle of

departure.Quadrant angle of de-

parture.lp corrected for E

TW

Wll

rpt

g

w

m

()

The following notations are used herein:

vV

Vw

Symbol Expression for

j Jump.

2

=



. DIRECT FIRE

Takes into account decreasein density of air as pro-j ectil e rises,

Takes into account increasedwind effect on projectile asit rises.

Takes into. account obliquepresentation of projectile tothe resistance of the air.

Feet.

Degrees, minutes, and tenthsof a minute.

Yards.Feet.

Expressed inInches.

Calibers.

r Reducing factor.

. fw Wind factor.

Yo Maximum ordinate oftrajectory.

Characteristic index.T H E GUN

The followingis quoted from Gunnery, of D.R.C.A., 1905:A Gun is a machine by means of which the force of expanding powder

gas is utilized for the purpose of propelling a projectile in a definite direction.It consists essentially of a metal tube, closed at one end, of sufficient strengthto resist the pressure of the gases, in which is placed a projectile designed tomove through the tube. The force of the expanding gases propels the pro-jectile through this tube causing it to start in its trajectory in a definitedirection.

The explosion of the powder gives rise. by its decomposition, to a largeamount of gas. which tends to expand in all directions and to occupy a spacemuch greater than that in which the powder was contained, and conse-quently exerts pressure in every direction.. The energy utilized in developing the force which propels the projectileIS heat and the heat carrier is gaseous, therefore a gun is the simplest form

Symbol Expression ford Diameter of bore or

caliber of projectile.n Radius of ogive. Also

twist of rifling.E Depression angle due

to height of site andcurvature.

R Horizontal range.

X Horizontal range.C Ballistic coefficient un-der range-table con-ditions. .

C1 Ballistic' coefficient un-der non-range-tableconditions.

c Coefficient of form.c')1 Standard atmospheric

density .. Act u a I atmosphericdensity.

fa Altitude factor.

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4DIRECT FIRE

of gas engine. The expam;ion of gases which develops the force that drivesthe projectile is entirely due to heat, for a gas at absolute zero temperature

would have no tendency to expand.The direction in which the projectile leaves the muzzle, is determined by

the pointing of the gun. For any particular gun the initial velocity withwhich the projectile starts in its path through the air is determined by thekind, weight, and condition of the powder used, the shape and dimensions ofthe powder grains, the density of loading, and the weight of the projectile.

The force of the powder gas acts upon the projectile while it is in thebore and for a short distance after it leaves the muzzle, that is while it is inthe powder blast. It is therefore manifest that a large amount of this forceis wasted, in consequence of the practical limitation in the length which canbe given to a gun. Experiments have shown that only about 1/8 of thetotal energy of the expanding gases is ever utilized in a gun of ordinarydimensions; this portion is termed the total work in the gun. .

A portion of this work is lost in heating the gun, and this loss of heat

reduces the amount available to prC?duceexpansion.

The total work in the gun is expended as follows:

In heating the gun.

In expanding the walls of the gun.In driving the products of combustion through the bore, and indriving out the column of air in front of the projectile.

In deforming the rotating band and in overcoming friction in the bore.

In giving rotation to the projectile.In accelerating the projectile, that is gradually increasing its velocity.

As the prime object is to give acceleration to the projectile, it is impor-tant to know how much of "the total work in the gun" can be utilized forthis purpose. Experiments have shown that between 80% and 90% of"the total work in the gun" is employed in accelerating the projectile.

If a charge of powder is burning in a rigid space, that is one that has afixed capacity, the expansive force of the gas will at any instant dependupon the amount of gas which has been formed and its mean temperature.As the amount of gas or its temperature increases, the expansive force willincrease. If now at any instant the space in which the powder is burningbe increased, the other conditions remaining the same, the expansive forcewill decrease. In a gun the space in which the powder is burning has afixed capacity until the shot starts, after which the space increases.

The pressure on the walls of the gun and on the base of the projectile

at any instant depends upon the expansive force of the gas at that instant.As long as the increase of expansive force, due to an increase of the amountof powder burned or to temperature, is greater than the decrease of expan-sive force due to an increase in the space it occupies, the pressure on the wallsof the gun and base of the projectile is incrensing. 'Vhen the decrease dueto the increase in the space,is greater than the increase due to the nmounl ofpowder burned or to temperature, the pressure falls off.

This is what happens in a gun; the pressure increases up to a certainamount and then begins to fall off. The greatest pressure is cnlled the

"maximu"m pressure" and this is the pressure indicated by the pressuregauge. This pressure musl not exceed that which it is considered safe to

use in the gun.The form and size of the grain should be so regulated as to produce the

"



DIRECT FIRE

greatest possible muzzle velocity with a maximum pressure not exceedingthat which is allowed for the gun.

It is sufficient to say here that within certain limits the larger the grainthe less will be the initial burning surface per pound of powder, and the lesswill be the maximum pressure produced for a given charge of po~der. Thiswill admit of using a larger charge of powder which will, as a rule, give agreater muzzle velocity. As the size of grain is increased the percentageof the grain which will be consumed while the projectile is in the bore isdecreased, leaving a certain portion of each grain unburned when the pro-jectile leaves the muzzle. \Ve will finally reach a point at which the weightof the unburned powder will equal the additional amount of powder which.has been put in the chamber and of course no further increase of velocitycan be obtained and if we continue to increase the size of grain without in-creasing the charge, the velocity will fall off. From which it is manifestthat for each class of gun there is a limit to the size of grain which must notbe exceeded. There is also a limit in the opposite direction and it is pos-sible to produce dangerous pressures in a gun by even a small charge ofpowder the grain of which is too small for the gun.

For this reason a powder should not be used in a gun for whtch it is potdesigned. If circumstances make it absolutely necessary to do so, use apowder having a grain which is larger than the normal for the gun.

The maximum pressure and the muzzle velocity are also affected bythe density of loading; it is important therefore that the projectile shouldalways be seated so that its base will be at the same distance from the faceof the breech, in order to insure obtaining uniform muzzle velocities.

The uniformity of action of the powder is also greatly dependent uponthe ignition and inflamation of the charge. By ignition is meant setting fireto the charge, and by inflamation the spread of the flame frum grain to grain,over the surface and into the perforations of the several grains. It is desir-able to produce as nearly a simultaneous ignition of the entire charge aspossible, in order to eliminate the variation in the rate of emission of gas,due to an irregular spread of the inflamation from grain to grain.

Black powder ignites very readily, but smokeless powder requires alarger amount of flame to insure instantaneous ignition; for this reason anigniting charge of black powder is attached to each section of a cartridgeof smokeless powder.

An extremely hard or very smooth grain is difficult to ignite. A chargecomposed of large grains inflames more readily than one composed of smallgrains, in. consequence of the size of the interstices. A grain with largeperforations inflames more readily than one with small perforations. With

multiperforated grains, the longer the grain the slower will be the inflama-tion with the same sized perforations.

It has been ascertained by experiment that, if a cartridge is muchshorter than the powder chamber, there is a liability to produce a motion ofor in the gas formed in the chamber, which materially increases the maxi-mum pressure on the breech. This motion has been generally considered tobe a "Wave Motion."

.An e~tremely long chamber renders simultaneous igniti~n and inflam-~atIon dIfficult and tends to produce so-called "wave motions." Hence,m order to contain the required amount of powder without giving it unduelength, the chamber is given a greater diameter than the main bore.

The rate of emission of gas from a burning grain is dependent upon

j



6 DIRECT FIRE

In making up cartridges care should be taken that the bags are of thesame dimensions, especially as to length. The length of the cartridgeshould not be less than nine-tenths of the length of the chamber.

THE TRAJECTORY

In order to reduce confusion of ideas to minimum, inthis article the several elements of the trajectory are stated to

he as follows:Trajectory.- The curve described by the center of gravityof the projectile in passing from the .muzzle of the gun to the

point of impact.Angle of departure.- The indination of the line of de-

parture to the line of shot. .Line of shot.- The line from the trunnions to the point of

impact.

(See Fig. 1.) ]' is the point of impact when the point of .impact is in the horizontal plane containing the axis of thetrunnions, sometimes called the initial plane. The value.described as the angle of departure in the range tables, andreferred to herein as cp , is represented by the angle A G' I '.As the gun is raised above the water level (the reference sur-face in coast artillery work) the point of impact lies in a planebelow the initial plane, at a point I. The line of shot noW

has a depression represented by the angle l'G'l, and, in orderto compensate for the change in the height of site, the piece

its rate of burning and the area of the burning surface. The rate of burningincreases with the pressure, and also probably with the temperature of the

inflamed gases.

As a rule the larger the grain the less will be the area of burning surfaceper pound of powder, and the slower will be the rate of emission of gas forthe same rate of burning, for any given charge.

Whatever may be the form of a grain there is always one dimensionthat when burned through, the entire grain is consumed. Thus with twotubular grains of the same length, one of the 72 inch in diameter and theother 1 inch in diameter, both having the same thickness of wall, say 1/8of an inch, it is manifest that as soon as the wall is burned through bothgrains are consumed. The larger of these grains would weigh about twiceas much as the other, and twice the amount of gas would be evolved. Thisdimension is called "the least dimension" of the grain, and som~times the"critical dimension" of the grain. In a multi-perforated grain the criticaldimension is the thickness of the web between the perforations.

For similar guns the "critical dimension" of the grain must be propor-

tional to the caliber of the gun.When a grain is said to be too large or too small for a gun, too quick or

too slow for a gun, it is the "critical demension" which is spoken of.Uniformity in the size, shape, density, surface smoothness, and tough-

ness of the grain is essential in order to obtain uniform results.* . * * * * * * * * * * *

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DIRECT FIRE 7

should be depressed by the same angular amount, I'G'IA G' B. In other words the angle of departure is keptsensibly constant, or A G'1' BG'I, if the principle of therigidity of the trajectory be accepted.

The principle of the rigidity of the trajectory assumesthat the trajectory preserves the same form, whether' thechord be horizontal, as G'I', or inclined like G'I. This isslightly erroneous, and its acceptance would cause the pro-jectile to range beyond 1', if the latter point were projectedvertically to the plane GI and designated I", notwithstandingthe fact that the measured .range GI" for which the piece

. would be laid, is slightly less than G'I". '*In order that the projectile may not range too far, the

quadrant angle of departure CPx must be slightly less than CPE;

that is, the correction to be applied to the range table angle

F~ /.1552

. of departure cP, in order to secure the quadrant angle of de-parture CPx, must be slightly greater than E. This constitutesthe essential difference between the values CPx and CPE.

If l' be projected vertically to the plane GI, the angleIG'I' so formed would represe~t the value E.

Quadrant angle of departure.- The angle BG'I', or the in-clination of the line of departure to the horizontal.

Line of departure.- The tangent to the trajectory at themuzzle, or the prolongation of the axis of the bore at theinstant the projectile leaves the muzzle ..

Angle of position (or depression)~- The inclination of theline of shot to the horizontal plane through the axis of thetrunnions. It is designated by E.

Pintle center.- The vertical axis about which the carriagerevolves.

Range.- The horizontal distance from the pintle center to~he point of impact; in Fig. 1, the distance GI. When the rangeIS expressed in feet, it is designated by X; when in yards, by R.

• See page 14, Ballistics, Part II, Major Alston Hamilton.

=

=

~



. ...

DIRECT FIRE

Muzzle velocity.-The velocity in feet per second withwhich the projectile leaves the muzzle in the direction repre-sented by the line of departure. Due to the action of gravity,

the projectile falls away from the line of departure.Angle oj jall.-The inclination of the tangent to the tra-jectory (at the point of impact) to the horizontal. The angleC'I'G' (w ) represents the -value given in the range table indegrees and minutes, or as a slope of fall. Under the servicecondition, the angle of fall is represented by the angle eIGw + E, and is represented herein as WE.

Ballistic coejJicient.~ The ballistic efficiency of the pro-

jectile. or its power to overcome the resistance of the atmos-phere and retain. its horizontal component of velocity. It

depends for its value on:(1) The ratio of the weight to the cross-sectional area;

wthat is, the "sectional density" represented by the ratio d2'In other words, the heavier the projectile per unit of area ex-posed to resistance, the more efficient will it be in retaining its

velocity.(2) The shape of the point. It is obvious that sharpening

the point will reduce the resistance encountered, will reducethe retardation, and will permit the projectile to retain itsvelocity the better. The value of sharpening the pointvaries with velocities, for a sharp point will obviously confermore advantage at 2000 feet per second than at 2 feet per

second.(3) The oblique presentation of the projectile to theatmosphere. The gyroscopic action of the rotating projectilecauses the point to precess so as to describe a small circle aboutthe tangent to the trajectory. The projectile is thereforenever exactly head-on to the resistance of the air; but, dueto the small curvature of direct fire trajectories, the preces-sion probably never describes. more than the first quarter

of the circle referred to above, so that the point is slightlyabove and to the right of the tangent to the trajectory. Thiscauses the projectile to be obliquely presented to the air andconsequently to drift to the right. It can also be observedthat the projectile will tend to "kite."

The factor i is called the characteristic index of the pro-jectile; it is determined experimentally; it embraces thevariation in retardation along the trajectory noted under (2)and (3); and it is placed in the equation to credit a projectile

\ I

=

-

-



DIRECT FIRE 9

(having a. given sectional density) with. a ballistic coefficientequivalent to the ballistic coefficient of the projectiles usedin the firings on which Tables I and II are based, and takes

the place of the expression ~: 10I}.gused in our service.The leading thought at present is to place the equation

. for the ballistic coefficient in the following form:

C i~2 (See Americana, 1912 edition.)

(See range tables for long pointed projectiles.)For the above expression, the wind is assumed to be zero

along the range, and the atmosphere normal with 78 % satur.ation with moisture, the average saturation along the Gulfand Atlantic coasts.

It is obvious that the ballistic efficiency of the projectilewill vary accordingly as there is a head or rear wind com-ponent, and with variations of atmospheric density. There-fore, while

wC id2

represents the equation for the ballistic coefficient underrange table conditions, the general expression is

01 W

C fw • T . id2

The values of i for the purposes indicated above havebeen determined for the computation of range tables for longpointed projectiles, as follows:

For 5-inch and 6-inch guns i .61For 8-inch, 10-inch, 12-inch, and 14-inch i .63For projectiles having a general design similar to those

used in making the firings on which our ballistic tables arebased (shot and shell for seacoast guns), i is a constant; forother types of projectiles (shot and shell for land. guns, andshrapnel) i is usually not constant, and is determined as afunction of the range by experimental firing.

Existing range tables for capped projectiles were. con-structed on the value of the ballistic coefficient expressed asfollows: .

wC =fa. -d 2

"'(c

The general expression becamew a

C fa • "'(cd2 • -+ .fw

..... iiiiiiiiiiiiii=:=====.:t:1 ==::::::::::::~~E

=

=

=

= =

=



10 DIRECT FIRE

The following dis.cussion of what the ballistic coefficientreally is, is taken from Exterior Ballistics (O'Hern).

4. Determination of the resistance of the air.-The relation betweenthe velocity of a projectile and the resistance opposed to its motion by theair has been the subject of numerous experiments. The usual methodhas been to determine the loss of velocity over a known path. By equatingthe corresponding loss of, energy with the product of the resistance timesthe path the value of the resistance has been determined.

Thus letw weight of projectile in lbs.m mass of projectile wig.d. diameter of projectile in inches.

Vi velocity of projectile at beginning of measured p'ath.V2 velocity of projectile at end of measured path.s distance in feet between points of measurement of Vi and V2.

p resistance of air in Ibs.

R* =.!!!!.... retardation due to air resistance.w

Then mv 2 mv 22_

2 2

(I)

WhenceR* (V 1

2-V 22 )/2s (2)

In order that the computed values of R* may be accurate the distance smust be so short that p may be considered as uniform over that path.

Early investigators, such as Robins in 1742 and Hutton in 1790, de-termined the values of VI and V2 by means of a ballistic pendulum placedat varying distances from the gun. Later investigators have used electro-

ballistic instruments, the improvements in which have steadily increasedthe accuracy of the determinations. The latest forms of instrumentsand methods of measuring velocities are described on pages 32-40, Lissak'sOrdnance and 'Gunnery, Much data has been secured concerning variousprojectiles under various atmospheric and other conditions. How theseresults have been standardized and made applicable to other projectilesand other atmospheric conditions will be explained in the discussion of

the so-called Ballistic Coefficient.5. Mayevski's formulas for the resistance of the air.-The values used

in our present ballistic tables were derived by General Mayevski fromKrupp firings made at Meppen about 1881. In expressing the relationbetween the resistanee of the air and the velocity of the projectile, GeneralMayev,ski assumed that the resistance encountered by a given projectileunder given atmospheric conditions varied as some power of the velocity,the law of retardation being expressed by the equation

R* Anv"C

(3)

In which R retardation to be determined experimentally as per

equation (2) .• Represented by r by Ingalls and elsewhere in this artic1e.-G. A. W.

= = = = = = = =

=

__ __

=

=

=



DIRECT FIRE 11

C ballistic coefficient varying inversely as the retardation. Anequation from which C can be calculated will be given later.

n some power of the velocity.An a corresponding constant, n and An to be determined from equa-

tion (3) after substituting the values of v, R, and C.The following values of An and n were thus determined by Mayevski

and by Zaboudski.

Velocities, Ls. n log A Velocities, Ls. n log A

Below 790 2 5.66989-10 1370 to 1800 2 6.11926-10

790 to 970 3 2.77344-10 1800 to 2600 1.7 7.09620-10

970 to 1230 5 6.80187-20 above 2600 1.55 7.60905-10

1230 to 1370 3 2.98090-10

It will be notE>dthat the values of the log An taken in connection withthe corresponding values of vn are such as to make the values of R plottedas a function of v form practically a smooth curve.

More recent firings by the Gavre Commission in France have shownthat the above values of An and n are not strictly accurate but no ballistictables of greater accuracy have yet been computed. It now appears thatthe law of the air resistance is intimately connected with the velocity of

sound and that'Newton's Law of the Square, is in the main true.6. The Ballistic CoeiJicient.-In Mayevski's expression for R*, equation(3), he assumed

C (4)

In which(~l standard density of air, that is with barometer 760 mm., ther-

mometer 15°C, relative humidity ~.a density of atmosphere at time of experiment.

c coefficient of form. A general expression for its value will bededuced later.

w weight of projectile in lhs.d diameter of projectile in inches.

The value of (~da for any weather conditions may be obtamed fromTable A, of this pamphlet, or from Table VI of Artillery Circular M, withthe barometer and the thermometer readings "as arguments.

It is evident from equation (3) that An vn

IS directly equal to R onlyWhen. C 1. That is, the tabulated values of An vn and their derivedfunctIOns are based on the retardation encountered by a projectile of onepound weight, since w 1; of one inch diampter, since d 1; of standardform, since c 1; moving in an atmosphere of standard density, sinceaI/a 1. It then becomes evident that C is the factor by means of which

. the :V~lues of Anvn are made applicable to projectiles and atmosphericcondI~IOns which differ from these standards. A thorough understandingof C IS of such fundamental importance that the meaning and use of thefactors already appearing and certain others to be introduced later willnow be fully consIdered. It will be noted that C varies inversely as the

* Represented by r by Ingalls and elsewhere in this articlf'.-G. A. W.

=

= =

=

=

= = =

=

= = =

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12 DIRECT FIRE

(5 )

retardation and hence measures the ability of the projectile to maintainits velocity. Since the weight of air to be dIsplaced by a projectile perunit of travel will vary directly as the density of the air and as the areaof cross-section of the projectile, the retardation will vary directly as thesefactors. We accordingly find a/OJ and d2 as inverse factors in the expression'for C. Every factor but d2 has disappeared from the ratio of the areas,rd12/ rd2, because the diameter of the standard projectile dI, was assumed

as one inch.Since the retardation of a projectile acted upon by a given force will

vary inversely as the weight of the projectile, w will enter as a direct factor

in the expression for C.The factor c, called the coefficient of form, is habitually determined

by experiment.The following formula gives its value for an ogival head

c 2k !4n -1n '\f 7

In whichn radius of ogive in c3libers.k a constant which is ordinarily unity, but which is determined

experimentally and thus compensates for peculiarities of conformationnot attributable to the radius of the ogive. For k 1 and n 2, thevalue in general use in the U. S. Service, c 1.

So when we say that under a given set of atmosphericconditions a certain projectile has a ballistic coefficient of 11,we mean that it will suffer loss of velocity at only one-elevenththe rate suffered by a projectile of standard form, one poundweight. and one inch caliber, under normal atmospheric con-

ditions.The ballistic coefficient of the standard projectile under

normal conditions is represented by the equationwC cd'!.

in which c is called the coefficient of form and is unity for anogive struck with a radius of two calibers. It relates merelyto the shape of the head of the projectile.

The values of c for ogives struck with dIfferent radii,expressed in calibers, are taken as follows from page 39,

Ballistics, Part II, and page 295, Notes on Ballistics, JOURNALU. S. ARTILLERY November-December, 1909, by ~lajorAlston Hamilton:

n 2 3 4 5. 6 7

c 1. 00 0.82 0.71 0 . 64 O. 58 O. 51These agree with the most recent experiments at Indian Head and at

Sandy Hook. .Our long-pointed caps are struck with a radius of ogive

of n 7 calibers; yet for the 12-inch projectile i 0.63. But.the hourrelet is separated from the base of the wind shield ogive

=

= =

= = =

=

= =

=

=



DIRECTFIRE 13

by an axial distance of about half a caliber. This and theoblique presentation, etc., so reduces the value expected fromthe point that the gain is limited to a projectile of standard

form having a radius of ogive of about five calibers, flying exactlyhead-on, and conforming to the laws on which Table II is based.See Chap. II, Ballistics, Part I; pages 187-189, and 223-

225, Artillery Circular N; and Introduction, Artillery CircularM.

It will be noticed that in some of the range tables com-

puted by the Ordnance Department the expression j~is used

instead of the factor i.Striking velocity.- The velocity of the projectile in feetper second in the direction of the line of impact, and at thepoint of impact.

Jump.-Either the difference between the sight elevationand the sight angle of departure, or the difference between thequadrant angle of elevation and the quadrant angle of de-parture. In Case I the piece is given an angle of elevation

with reference to the line of sight; in Case II and Case IIIthe piece is given an angle of elevation with reference to thehorizontal; in either case the shock of discharge may changethe inclinatio"n of the axis of the bore at the instant of de-parture from the inclination at which the piece was laid.~his change in inclination is called jUlllp. The jump may be~Ither positive (increasing the inclination) or negative (decreas-mg the inclination), or may be zero (in which case the inclina-

tion is unchanged), "depending upon" the kind of carriage andplatform employed, the length of the bore, * * *and the angleof elevation." (For a study of jump, see JOURNAL U. S.ARTILLERY,Vol. 36, Sept.-Oct., 1911, page 176; Vol. 35, Jan.-Feb., 1911, pages 63-75.)

Angle of elevation.- The .inclination of the axis of the boret~ the line of sight, measured in a vertical plane, when thepIece is laid. Often called sight elevation.

. Quadrant angle of elevation.- The inclination of the axisof the bore to the horizontal, when the piece is laid.

Sight angle of departure.- The inclination of the line ofdeparture to the line of sight, measured in a vertical plane.

Jump AG'BAngle of departure IG'BAngle of elevation, or sight elevation TG' ASight angle of departure TG'B

_______________ S!l!!lJlllJl!l!lllJll!llllll---.-II!IlIII!I.-lIIiiIiiiiiiiiiiij;j

-

= = = =



14 DIRECT FIRE

Quadrant angle of departure CG' BQuadrant angle of elevation CG' A

Line of impact.- The tangent to the trajectory at the

point of impact; or, the direction of motion of the projectileat the instant of impact.

Angle of impact.- The inclination of the line of impact tothe surface hit at the point of impact.

Angle of incidence.- The inclination of the line of impactto Ule normal to the surface hit at the point of impact.

These last two angles are of interest chiefly in the studyof armor attack and the computations of perforations to be

.expected.

AKisor G'

7;-(/1111/0110$ :III

I

1663

Table II.-For a thorough understanding of its signifi~cance, consult Artillery Circulars, M and N. It consists of anumber of tables, each table being for a certain muzzle ve~

locity, and designed to display certain computed secondaryfunctions in such a manner as to be most convenient forballistic computations. Having given a horizontal distance X

(in feet), and a given C, and a given V, for the ratio (= Z)

there can be found, of chief interest, the value A ( sinC2lfJ

)

for the given muzzle velocity. Hence from A we c~n deter-mine cp, the required angle of departure. Other elements ofthe trajectory can likewise be computed from the formulregiven later on, which show the relation of the element that'we desire to compute to the proper function to be found

opposite the ratio Z ~. For convenience, an abridged

Table II is appended hereto, including the values of A forV 2150, V 2250, and V 2350 as determined by doubleinterpolation frem the 21CO, 22CO, 2300, and 2400 f.s. tables.

= =

~

~

=

~=

= = =



DIRECT FIRE 15

There are also appended the tables for muzzle velocities from2000 to 3000 inclusive for values of Z from 100 to 7000.

In the range tables for capped projectiles it is seen thatthe values of C vary with the range. These values are basedon assumptions which are no longer considered to representthe best practice; but it is sufficiently exact to take the nearesttabular value from the range table for any range considered.

Then the value of C l (for the day) . fw' C (range table

value).

Problem 1

Thus for the 12-inch gun and capped projectile, range8000, barometer 29", thermometer 40°F., and a range' com-ponent of wind 20 miles per hour against the projectile, wehave:

From Table A,Atmosphere factor .98

2lV T5/4Wind factor, fw 1:f:: X

(The positive sign is used when the wind is ac'celerating, andthe negative si~n when the wind is retarding.)

log 2 W x log 40 1.602065/4 log T 5/4 log 13.06 1.39493

colog X colog 24000 5.61979 -10

log .04138 8.61678fw' 1 .04= .96

(Range table) log C 0.90i)71

I()l

og log .98 9.99122 -10

log fw log .96 ,;, 9.98227 -10

log C 1 (for the day) 0.88320 log 7.642For the range tables for the long pointed projectiles it is

seen that the value of C is taken as constant throughout therange. The expression for C under range table conditions is

C i~2'

Problem 2

. Ass~ming the same range, atmosphere, and wind as inthe preVIOUS problem, we have for the 12-inch gun and longpointed projectile:

== ~

=

=

= = = =

= =

= = -

= ~ = =

=

= =

=





DIRECT FIRE 17

HAMILTON'S FORMULlE

which reduces to,

t =Pv~

in which t is the thickness of K.C. armor that a capped A.Pshot is to be expected to perforate at normal impact. P v is afunction of v to be taken from Table P which gives values of P

v

with v as an argument; wand d are, respectively, the weightof the A.P. shot in pounds and its diameter in inches.

Wind deflection (degrees) ~z Dw

0

in which Wy is the cross-component of the wind In miles perhour, and D 0 is the wind deflection function to be taken fron:Table D. w

Drift (degrees) d3

(1 K) (fP + w) sec fPwn

in which K is the drift constant, Ii the number of caliberswhich the projectile would have to move along the bore inorder to make one turn at the final rate of twist. n is or-dinarily 25 for direct fire guns of high power in our service,and the twist is to the right. For long pointed projectiles! { = .75. .

4s regards the precision to be attempted in the com-Putations, t~e following is quoted from page 3, Artillery Cir-CUlar N~ Senes of 1893, by Colonel James M. Ingalls, U. S. A.:. Logarithms are habitually employed in making the numerical computa.

hons, as it is believed that by their use a considerable saving of time andlabor is effected and the liability of error reduced to a minimum. In theabsence of a table of logarithms, however, most of 'the examples can bew~rked out by the four fundamental rules of arithmetic. Five place log-anthms are sufficient for the accurate solution of all gunnery problems, andfour place logarithms can often be used to advantage.

ApPLICATIONS OF THE FORMUL.lE

TARGET IN THE INITIAL PLANE

Problem 3Given R, C, and V, to determine fP.

R 8000; C (1.07168); V 2250.See Table II, appended.Z=X

C

iiiMiii m aLa

~

=

= - -

= = =



18 DIRECT FIRE

log A 8.18893 -10log C 1.07168

log sin 2rp = 9.260612rp 10° 30'

rp 5° 15'

which checks closely enough with the range table for the 12-inch gun and long pointed projectile.

log X 24000 4.38021colog C 8.92832 -10

log Z = 3.30853Z= 203535

For V 2250 and Z 2035, A .01513 + 100 X 91

.01545

Sin 2rp AC

Problem 4Given R, C, and V, to determine rp.

R 8000; C (1.07168); V 2230.Rand C being the same as in problem13, we get

Z 2035.

(V. 2200) A .01585 + 1~50X 95 .01618

(V 2250) A (see Prob. 3) .01545; Av

73(V 2230) A .01618 + ~gX (-73) .01574

Sin 2rp' = AClog A 8.19700 -10log C 1.07168

log sin 2rp' = 9.26868 -102rp'

10° 41'.9rp' 5° 21'

We see that 6' greater elevation is required than when

V 2250 f.s.

Problem 5

Given C, V, and rp, to determine R.

C (1.07168); V 2250; rp 6° 4'Sin 2rp AC

= =

= =

= = =

= = =

=

= = =

=

= = = = =

=

-= = =

= =

= =

=

= = = =



DIRECT FIRE 19

(V =2300) log C' =4.9032 -10

(V=2250) log C'=4.9032-10

XZC

log sin 2~ log sin 12° 8' 9.32260 -10colog C 8.92832 -10

log A 8.25092 -10A .01782

'. 86'For (V 2250) and A .01782, .z 2200 + 94 X 1?0

2291.5X

Z = c

log Z log 2291.5 3.36011log C 1.07168

log X 4.43179X 27027R 9009 yds.

Problem 6

Given V, ~, and R, to determine C.V 2250; 5° 49'; R 8400.C' = sin 2

X

log sin 2~ log sin 11° 38' 9.30459 -10colog X colog 25200 5.59860 -10

log C' 4.90319 -10

(V =2200) log C' =4.9032 -10 Z=2100 + ~ X 100= 2105A =959

Z =3000 + ~~X 100 =3064

Z 2105 + 1~~ X 959 2585

log X 4.40140colog Z colog 2585 6.58923 -10

log C .99063which is practically the log C for the lO-inch, long-pointedprojectile.

Problem 7

Given ~, R, an~ C, to determineV.

=-

= = =

= =

= = = =

= = =

= = =

= ~ = = ~

= = = =

=

= =

= = =

=



20 DIRECT FIRE

q; 6° 8', R 9000, C (1.07168)

Z=~

log X log 27000 4.43136colog C 8.92832 10

log Z 3.35968Z 2289

Sin 2q; AClog sin 2'P log sin 12° 16' 9.32728 -10

colog C 8.92832

log A 8.25560 -10A .01801

\Vith Z as an argument, we look through Table II to findthe two successive velocities whose A's for Z 2289 bracketthe value A .01801. Then we interpolate for the velocity.

89(V=2200) Z=2289 A =.01776+ 100 X99=.01864

A -.00084

(V=2250) Z=2289 A =.01696+1~90 X94=.01780

V 2200 + .01801 .01864 X 50 2200 + 63 X 50.00084 -84

2238 f.s.

Problem 8The following is quoted, with a few alterations, from

lVotes on Ballistics, by Major Alston Hamilton (pages 259 and260, JOURNAL U. S. ARTILLERY, Nov ..Dec., 1909).

In dealing with the jump of the piece it is necessary to find how farthe trajectory should naturally fall away from the line of departure, due tothe action of gravity and to the vertical component of the air resistance.

This is done as follows:

y (ac )-'- tan cp' 1 -.--x sm 2cp'

or, aCx tan cp'

y x tan cp . 2sm cp

That IS. aCx

x tan cp' y.. 2 cos 2cp'

and since for this purpose, cos 2 cp' may be taken 3S unity for direct fire, we

have

= = =

= = = -

= =

= = =

=

= =

== =

=

= - = - = -

= -

= - '

- = --



DIRECT FIRE

Natural drop a~x ft.

M aC 2 Z ft.6aC2 z inches.

21

* * * * * * * * * * *

The manner of solving this problem in general will be evident from theillustrntion given below: '.'

Given V ;, 2234; barometer, 30.31; thermometer, 37°; wind velocity,12 miles per hour; azimuth of wind, 128°; azimuth of fire, 316°; quadrantelevation, Q 10° 27'; drop ofprojectile from line of bore sights, measuredon jump screen, 23 in. Deviation on screen, 0.0 in. Measured range~

14,091 yards; measured deflection, 0°.44 right; time of flight (observed)23.13 seconds; height of tide, h, 2.5 feet; * * * distance, D, from muzzleto screen, 366.2 feet; distance, b, from axis of trunnions to muzzle of piece,24.1 feet; D + b, 390.3 feet; range in feet, 42,273; 12-inch B.L.R.; weight ofprojectile, 1046 lbs.; shot [long pointed]; radius of ogive, 7 calibers'; twist;1 turn in 25 calibers. * ' , .

Determine i and K.

From Table C, assume

c .54From Table A,

~l .933o

For the short distance to the screen assume fw 1.

a w 1046C -} . C d2 .933 0.54 X 144 12.55

N ow for the screen'X D 366.2

Hence

X z 366.2 29C 12.55

To find a we have, with z 29 and V 2234

z = 29

(V = 2200) a .000194 .000008(V 2250) a .000186

(V 2234) a ..000194 }~- X .000008 .000188

Hence from natural drop 6'aC2z inches,

1 • >II While f~om this ,Point on, the general method employed in this prob-eI!lIs that Major HamIlton employed in the reference cited yet it is not

Pbrmted in the smaller type because the agreement is not s~ close as haseen the case up to this point.

=

=

=

=

= = =

= =

= = =

= =

~ -= =

= = -

=



22 DIRECT FIRE

log 6 0.77815log a 6.27416 10

2 log C 2.19728

log z 1.46240

A.02120.02905

785

267.02387

log natural drop 0.71199Natural drop 5.15 inches. .

The measured drop is 23 inches; hence the' drop due to

negative jump is 23 5.2 17.7 inches.N ow this distance is measured on a screen at a distance

from the axis of the. trunnions which equals D + b, or 390.3

feet, or 4684 inches. Then, since a radian is 34i38', we have

Jump ~~8~X 3438' 13'

Hence the quadrant angle of departure, cp,., is the quad-

rant elevation 10° 27' 13' 10° 14'.To find E (see Fig. 2):The height of. the trunnions above mean low water was

24.7 feet; the height of the tide was 2.5 'feet; the curvatureof the earth at this range (see Table K) is 42.8 feet.

H. _ 3438' 42.8 + 24.7 2.5

ence E .' 42,273

3438 . 42~;73 5' .3

Hence cp ' cpx + E 10° 14' + 5' .3 10° 19' .32cp' 20°'38' .6

We now know Y, cp', and X, and can calculate

C' _ sin 2cp'

X

with which to enter Table II, Artillery Circular M, or TableB, appended, and find thence, by interpolation. the value of C J

(for the day), as follows:log sin 2cp' log 20° 38'.6 9.54722 -10

colog X colog 42,273 5.37394 -10log C' 4.92116 -10

Log C' to four places = 4.9212 -10

With this value fA log C' we getyZ

2200 25402300 3482

A (100) 942A 34 3202234 2860

= = -

=

= = =

- =

= - = -

- =

--

= =

= = = =

-

= =

=

=

=



xz sin 2 cp. A C. C

log X 4.62606 log sin 2cp 9.54722 -10

colog Z 6.54364 -10 colog A 1.62215log C 1 1.16970 log C 1 1.16937

Hence it is seen that the values of log C 1 agree veryClosely as secured through different methods of interpolation.The corresponding values of C 1 are 14.781 and 14.77.

Take the mean value, C 1 = 14.776 (for the day). Butt~is C 1 "= the range table C multiplied by the atmosphere and

WInd factors. Designating the C for the day as Ct,

C 1 xfw X C

To find !w ' note that2 W x T5/4

fw I::!: X

The negative sign is used for a head wind and the positive

sign:for a wind from the rear. "In''-this case the azimuth of fue is 316° and the azimuthfrom which the wind is blowing is 128°. Hence the wind isblowing from the right rear and makes an angle of 360°316° + 128° 172° with the plane of fire, or an acute angleof 8°. The range component is therefore \V cos 8°, and the~ross ..component from the right is W sin 8° and tends to dimin-Ish the drift.

DIRECT FIRE23

log W y 0.2228" W y -1.670

Wind 12 miles per hour;log W 1.0792 log W 1.0792

log cas 8° 9.9958 log sin 8° 9.1436

log W x 1.0750W x 11.885

To find fw' since T 23.13log W x 1.0750

5/4 log T 1.7052log 2 = 0.3010

colog X 5.3739

log .029 8.4551 10

fw 1 + .029 1.029

(Table A) a; .933

= - =

= =

= = = =

= ~

=

-=

= =

= = = =

= =

= = =

=

= -

= =

=



24

c=

DIRECT FIRE

C l

01T .fw

log C 1 log 14.776 1.16955I 0'1

CO og " colog .933 0.03012o

colog fw colog 1.029 9.98759 -10

log C under range table conditions 1.18726

C w . wi d2 or l C d2

log W log 1046 3.01953colog C 8.81274 -10

colog d2 colog 144 7.84164 -10

log i 9.67391 -10l .47

(funPhn~ or /Jirecfion

Pbneor Shof-

FIG. 3.

470rget

POlnt-OT /mppcf

1554

Lateral Deviations or Deflections

The line of direction is the line from the pintle center tothe aiming point, or to the center of the target, at the instantthe shot strikes.

The plane of direction is the vertical plane containing theline of direction.

The plane of departure is the vertical plane containing theline of departure.

The plane of sight is the vertical plane containing the lineof sight at the instant of departure. . Range firings are or-dinarily conducted at a fixed target, or at an aiming point.

In this case the plane of departure coincides with the planeof sight. That is, the piece is laid on the target withoutlateral corrections. Then the fall of the shot to the right orleft of the plane of direction is observed, and is expressed inangular measure. The observed angular horizontal divergenceof the point of im'pact is' modified by the divergence causedby the lateral component of the wind. This gives the drift,and ultimately the constant for the projectile.

To find the drift constant, K, the observed directionmust be corrected for the cross wind component..

= = = =

= =

=

= = = =

= = =

= =



DIRECT FIRE 25

W~. 1.67 5/3

The value of the wind deflection is,

Wind deflection (degrees) WVZ X D wo

N ow for V 2234, from Table. D we .have for Z 2860

Dwo 0°.0061

I-Ience,

Wind deflection 5/3 X ;~~~ X .0061 .013

The observed deflection was 0°.44 right. This increased by .013becomes 0.45. \Ve have, then, from the formula for drift

DO (1 K) .!!- (cp' + w')s~c cp' (*)wn

0.45 (1-J\")10lZ~25 (cp'+w')sec cp '

It is necessary to find w'. We have

V 2234Z = 2860cp' 10° 14'

Hence for Z 2860 from Table B,

V log B'2200 .1073 -102300 .1063

342234 .1073 -100 X 10 .1070

tan w' B' tan cp '

log tan cp' log tan 10° 14' 9.25654-10log B' 0.10700

log tan w' 9.36354 -10w' 13° 0'.3 13°

cp ' 10°.23cp' + w' 23°.23

log sec cp' .00697Hence

(I-I\") 0.45 1046 X2523.23 sec cp" 1728

* From Hamilton's Ballistics, Part I, page 33.

= - = ~

=

= =

=

= - = -

= -

=

=

=

= ~

=

=

= = =

= = =

= = =



26 DIRECT FIRE

log 0.45 9.65321 -10colog 23.23 8.63395 -10colog see cp' 9.99303 -10

log 1046.0 3.01953log 25.0 1.39794

colog 1728.0 6.76246 -10

log (I-K) 9.46012-101 !{ .2385

K .712To summarize:

i .47K .712j* -'13'

THE RANGE TABLE

The range table for a direct fire. gun and its type pro-_jectile exhibits. principally the angle of departure as a function

of the range under an assumed set of conditions called "range-"table conditions"; these are: .1(u) That the gun and target are on the same level.(b) A constant value for the ballistic coefficient. . .(c) Law of retardation as assumed in Table II.

(d) Normal atmosphere; i.e.,~l 1, atmosphere 78%

saturated with moisture.(e) No wind.(f) Normal muzzle velocity.The elements of the trajectory shown as functions of the

range are:1. Time of flight.2. Angle (and slope) 0 fall.3. Maximum ordinate.4. Striking velocity.5. Perforation of Krupp armor.6, Deviation from the plane of fire due to: (1) drift; and

(2) component of wind of 10 miles per hour perpendicular tothe plane of fire.

COMPUTATION OF A RANGE TABLE

(NOTE.- The range table value of C here given does not

correspond with that to be found in the range table for the .* Jump.

j

= = = = = =

= - =

=



DIRECT FIRE 27

12-inch gun and long-pointed projectile. One who desires togain facility in the use of Table II may assume certain values inthe range table and solve for others, thus framing his own prob-

lems and checking his results with values in the range table.)Assume the values determined in Problem 8. We have

found that log C (range-table conditions) 1.18726. Com~pute the line of the range table for 10,000 yards:

log X log 30,000 4.47712colog C 8.81274 -10

log Z 3.28986Z 1949

(See Tables B and II.)(V 2250) Z 1949, A .01468

Z 1949V

22002300

log B'.0725.0716

u17121796

T'1.005

.960

.1441 3508 1.9652250 .0721 1754 .983

(See Table D.) V =2250, Z 1949, Dwo =.0059To find cp, sin 2cp AC

log A log .01468 8.16672 -10. log C 1.18726

log sin 2cp 9.35398 -102cp 13° 3'.5

cp 6° 31'.8

To find T, T CT'cos cp

log C 1.18726log T' log .983 9.99255 -10

colog cos cp colog cos 6° 31 '.8 0.00283

;

To find Yo, Yo

log T 1.18264. T 15.23 seconds

4.05T22 log T' 2.36528

log 4.05 0.60746

log Yo 2.97274Yo =.939 feet

=

= = =

= =

= = = =

= =

= = =

= = =

=

= = =

= =

= =

= = =

=



28 DIRECT FIRE

To find w, tan w B" tan cp

10gB' 0.07210log tan cp log tan 6° 31'.'8 9.05868 -10

log tan w 9.13078 -10w 7° 41'.9

To find sJope- of fall, 1 tan w. . x

log 1 0.00000colog tan'w 0.86922

log x 0.86922

x 7.4

d u cos cpTo fin vw, Vwcos w-

log u log 1754 3.24403log cos cp log cos 6° 31'.8 9.99717 -10

colog cos w colog cos 7° 41'.8 0.00394

log Vw 3.24514Vw 1758 f.s.

To find the perforation (see Table P),

t Py~~- 1.395 X ~~4~~13.0 inches.

For perforation at 30° with the normal, subtract 8 % fromthe above. (See Table Q.)

t30 13 X .92 11.96 inches

To find the deflection for a 10-mile wind we have,

Deflection (degrees) I~Z . "DvrO

. 1~~;~9 X .0059 .0511

To find the drift,

d3

Drift (degrees) (1 -!{) -( cp + w) see cpwn

.289 X. 10~~~25 X (6°.5 + 7°.7) X .9~35 0°.273I

Problem 9

Drift of Capped Projectiles; i.e., those not fitted with longpoints. *

* See Hamilton's Ballistics, Part I, Page 33.

IJ

= =

= =

= = =

= =

=

= = ---

= = = =

= =

= =

= = =

= =

=

=

=

= =

1

I

~



DIRECT FIRE 29

"Take K for shot and steel shell as 0.75; for C. 1. shell as0.80." .

Capped C. I . shell are being fired as trial shots from a

lO-inch B.L.R. w 604; ~1 1; lV 0; range to center

of impact,. 8000 yards; assume jump as ::f:: O. Assume thepiece to be laid accurately at the aiming point.

(a) If the gun pointer has his sight set at 3.00 andrnoves the vertical wire to each splash, what should the sightsetting be?

(b) What deflection should be computed by the de-

flection board fitted with a leaf range scale for steel pro-jectiles?

(c) What is the error at this range to be expected fromthe deflection board?

(See range table for 10-inch gun, and capped projectile.)a b

C. 1. Shell Steellog ( l -K) log .20 9.3010 log .25 9.3979

log d3 log 1000 3.0000 3.0000log (<p+w) log 14.4 1.1584. 1.1584log sec <p log sec 6°= .0025 .0025

colog w colog 604 7.2190 7.2190colog n colog 25 8.6021 8.6021

log drift 9.2829 9.3799drift 0°.190 0°.240

(a) The reading of the sight will be 3.00 0.19 2.81.This is the correct setting of the sight.

(b) The deflection computed will be 3.00 .24 2.76.(c) This would cause the projectile to fall 2.81 2.76

0°.05 to the left, and would necessitate a flat correction of +0°.05.

THE GRADUATION OF THE RANGE SCALE

At the present* writing the decision has been reached onthe proposition of correcting for jump. It is finally deemedpracticable, and the correction therefor will be incorporated inthe graduation of the range scale. A method here outlinedpermits a battery commander to COlllpute a table expressingthe Range-Quadrant-angle-of-departure relation. Any ma-terial discrepancy between his table and the elevation table

* November, 1914.

=

= - = =

= = = = = = =

= = =

= =

= =

- =

- = - =



30 DIRECT FIRE

furnished the battery for graduating the range scale by theclinometer,may be attributed to the correction for jump. I tis of interest to know that such a correction is to be applied.

\\Then solving for range corrections for abnormal condi-tions of atmosphere, wind, velocity, and tide, the jump Cor-rection is only incidental. We are no longer concerneddirectly with the quadrant angle of elevation (<t'I), but onlywith the quadrant m ~-f,lef departure (<t'x). Therefore, whetherjump be considered or not, we are in a position to computerange corrections by the table expressing the Range-<t'x re-

lation.To obtain th~ quadrant angle of departure for service

conditions, it is necessary to correct for such errors as areintroduced by the following assumption:

Gun and target on the same level.

To obtain the quadrant elevation it is also necessary tocorrect the quadrant angle of departure for the jump of the

carriag~.The elevation table will exhibit the angles of quadrant

elevation as a function of the range at suitable intervals, andis computed from the range table angles of departure by thefollowing formulre:

<t'i <t'x + Ll<t'J<t'x <t' + Ll<t'K + Ll<t'h

In which<t'l = quadrant elevation.<t'x quadrant angle of departure.<t' = range-table angle of departure.

AcpJ = the correction for jump taken as a function of <t'x.Ll<t'K = the correction for curvature.Ll<t'h the correction for height of site.

Angle of depression due to height of site, h in feet, assum_ing the principle of rigidity of the trajectory:

(1146) .6.<t'h= h mInutes (1)

Correction for curvature of the earth:R .

6. <t'K 4000 mInutes (2)

Correction for height of site without the assumption ofthe principle of rigidity of the trajectory:

= =

=

=

~

=



DIRECT FIRE

Ah 1134 + .005R .. tL.1«'h R mlnu es

31

(3)

This last formula is convenient where a number 'of eleva-tion tables for batteries on different heights of site are to becomputed. The values of the fraction for successive rangeshave been computed and tabulated.

Ll«'J is determined separately for each type gun and mount.

The procedure for computing the Range-«'l relation for.

a gun will be possible as follows:Select 9000 yards for an example for the 12-inch gun and

long-pointed projectile; height of site, 200 feet; M. V.,.2250f.s.

1. From the range table (Form 1014), «' 6° 4'.2. The correction for curvature of the earth 2'.25.3. The correction for height of site 26'.2. [Equa-

tion (3).] .Hence the quadrant angle of departure for the gun is,

i(Jx 6°4' 2'.3 26'.2 5°35'.5.If the line of departure makes an angle of 5°35'.5 with the

horizontal plane through the trunnions, we can expect theshot to fall at a horizontal distance of 9000 yards on the sur-face of the water, under range-table conditions.

5. Assume the verity of the jump curve appended tothis article. At an angle of 5°35'.5 the jump is seen to be about

+ 1'.3. That is, the gun increases in elevation that amountat the instant of dis harge .Therefore, the correction at that elevation is 1'.3.6. 5°35'.5 1'.3 5°34'.2.

If now the gun is fired with an elevation, «'1 of 5°34'.2,we can expect the quadrant angle of departure to be 5°35'.5.

To graduate the range scale, the gun is given an elevation«'1 5°34~.2 and opposite the index is cut and numbered the9000-yard range graduation.

At present, and until the range drums require regradua-tion for the long pointed projectiles, the graduation of the rangescales is as follows:

1. Take «' from the range table for the capped projectile.2. Determine Efrom Artillery Note No. 29. .3. «' E «'E.

4. Set the gun by the clinometer to «'E, and oppositethe index graduate the range drum.

=

= = -

= -

= - - =

-- =

=

- =



32 DIRECT FIRE

A METHOD FOR THE DETERMINATION OF DATAFOR AN ELEVATION TABLE

CPI CPx + .:1CPJ

<PK cP + .:1<PK + .:1<Ph

Arps for illustration, will be taken from the appended jumpcurve for disappearing carriages. Jump varies with the quad-rant elel'ation.

The following table has been computed.

F = 1134 + .005R .:1<PK = ~R 4000

Range F .:1<PK Range F .:1<PKminutes minutes.

1000 1.1390 .25 10000 .1184 2.50200 .9500 .30 200 .1162 2.55400 .8150 .35 400 .1140 2.60600 .7138.40 600 .1120 2.65800 .6350 .45 800 .1100 2.702000 .5720 .50 11000 .1081 2.75200 .5205 .55 200 .1063 2.80400 .4775 .60 400 .1045 2.85600 .4412 .65 600 .1028 2.90800 .4100 .70 800 .1011 2.953000 .3830 .75 12000 .0995 3.00200 . .3594 .80 200 .0980 3.05400 .3385 .85 400 .0965

3.10600 .3200 .90 600 .0950 3.15800 .3034 .95 800 .0936 3.204000 .2885 1.00 13000 .0922 3.25200 .2750 1.05 200 .0909 3.30400 .2627 1.10 400 .0896 3.35600 .2515 1.15 600 .0884 3.40800 .2413 1.20 800 .0872 3.455000 .2318 1.25 14000 .0860 3.50200 .2231 1.30 200 .0849 3.55400 .2150 .1.35 400 .0838 3.60600 .2075 1.40 600 .0827 3.65800 .2005 1.45 800 .0816 3.706000 .1940 1.50 15000 .0806 3.75200 .1879 1.55 200 .0796 3.80400 .1822 1.60 400 .0786 3.85600 .1768 1.65 600 .0777 3.90800 .1718 1.70 800 .0768 3.957000 .1670 1.75 16000 .0759 4.00200 .1625 1.80 200 .0750 4.05400 • .1582 1.85 400 .0741 4.10600 .1542

I1.90 600 .0733 4.15

800 .1504 1.95 800 .0725 4.20

= =



. DIRECT FIRE 33

Range F AcpK : Range F AcpK

minutes minutes

8000 .1468 2.00 17000 .0717 4.25200 .1433 2.05 200 .0709 4.30400 .1400 2.10 400 .0701 4.35600 .1369 2.15 -600 .0694 4.40800 .1339 2.20 800 .0687 4.45

9000 .1310 2.25 18000 .0680 4.50200 .1283 2.30 200 .0673 4.55400 .1256 2.35 400 .0666 4.60

600 .1231 2.40 600.0659 4.65

800 .1207 2.45 800 .0653 4.7010000 .1184 2.50 19000 .0647 4.75

Problem 10

Assume a 12-inch gun, 200 ft. above mean low water, andfiring long pointed projectiles. Compute an_ elevation table

for ranges of 8000 to 10,000 yards inclusive. (See page 34.)Due to the difficulty of reading the jump curve, it is well

to check the column CPt by differences. It is seen that thesecond differences in column 3 are inconsistent and shouldprobably be' all equal to +0'.1. Working conversely fromcolumn 4, we get the adjusted first differences in column 5.and adjusted values of CPI in column 6 for further use.

. al a2Representmg _. by A, and -2 2 by B, we have, generally, whatever them m

value of m,

~l A (m -1 ) B

~2 A (m-3) B ~l + 2B

~3 A (m-5)B ~2 +2B

~4 A (m -7 ) B ~3 + 2B

~6 A (m -q ) B ~4 + 2B

etc., etc., etc., ad libitum, according to the values of m.

For m 3; A oI/3, and B 02/18

m 4; A 01/4, and B 02/32m 5; A oI/5, and B 02/50m 10; A oI/10, and B 02/200

For our present work we wish to interpolate values of CPI

200for every 20 yards. Therefore we tnke m 20 10.

= -

= - = = - = = - = = - =

= = = = = = = = = = = =

= =



:34 DIRECT. FIRE

Range ({J ti.({JK ti. ({ Jh ({Jx J ({Jlyds. deg. min. minutes minutes deg. min. minutes deg. min.

8000 5 15.2 -2.00 -29.36 443.8 -2.0 441.800 524.7 -2.05 -28.66 454.0 -1.8 452.200 534.2 -2.10 -28.00 504.2 -1.7 502.500 544.1 -2.15 -27.38 5 14.6 -1.6 5 13.000 554.0 -2.20 -26.78 525.0 -1.5 523.5000 604.0 -2.25 -26.20 535.5 -1.3 534.200 6 14.1 -2.30 -25.66 546.1 -1.2 544.9006 24.4 -2.35 -25.12 556.9 -1.0 555.900 634.7 -2.40 -24.62 607.7 -0.9 606.800 645.1 -2.45 -24.14 6 18.5 -0.7 6 17.80000 655.6 -2.50 -23.68 6 29.4 -0.5 f) 28.9

Column 1 2 3 4 .5 6Range ({Jl (h 02 (02) (01) ({Jlds.. deg. min. min. min. min. min. deg. min.8000 441.8 10.4 -0.1 +0.1 10.3 441 <300 452.2 10.3 +0.2 +0.1 10.4 4 52.100 502.5 10.5 +0.0 +0.1 10.5 502.500 5 13.0 10.5 +0.3 +0.1 10.6 513.000 523.5 10.8 -0.2 +0.1 10.7 323.6000 534.3 10.6

+0.4 +0.1 10.8 534.300 544.9 11.0 -0.1 +0.1 109 545.100 555.9 10.9 +0.1 +0.1 11.0 5 55.800 606.8 11.0 +0.1 +0.1 11.1 606.900 6 17.8 11.1 -- - --- 11.2 6 17.90000 628.9 --- -- -- - -- 629.0

From 8000 to 8200,

A -.!!.- = 10.3 1 03' B ch 0.1. - m 10 - " = 2m2 = 200 = .0005

ti.1 A - (m-1) B 1.03 9 X .0005 1.0255ti.2 = ti.1+2B 1.0255+.001 = 1.0265 ti.

7= 1.0315

ti.a = 1.0265+.001 = 1.0275 ti.s = 1.0325ti.. = 1.0275+.001 = 1.0285 ti.

9= 1.0335

ti.1i = 1.0285+.001 = 1.0295 ti.10

= 1.0345ti.6 = 1.0295+.001 = 1.0305, etc ..

Hence.

- -

-

= ~ - = ~



DIRECT FIRE 35-

R ~1 . ~ldeg. min. deg. min.

8000 4 41.8 4 41.820 4 41.8+Al =4° 42'.8255 4 42.840 4 42.8255+A2=4° 43'.8520 4 43.960 4 43.8520'+Aa =4° 44'.8795 4 44.980 4 44.8795+A4=4° 45'.9070 4 45~9

8100 4 45.9070+As=4° 46'.9365 4 46.920 4 46.9365+A6 =4° 47'.9670 4 48.040 4 47.-9670+A7=4° 48'.9985 4 49.0

60 4 48.9985+As=4° 50'.0320 4 50.080 4 50.0320+A9=4° 51'.0655 4 51.18200 4 51.0655+AIO=4° 52'.1000 4 52.1

Determine a new A. and a new B for 8200 to 8400, anddetermine ~l for every 20 yards, etc.

For a further discussion see Appendix III, Hamilton'sBallistics, Part I.

Following such a system of interpolation, the followingportion of an elevation table has been computed:

ELEVATION TABLE(Columns 1 and 5)

R-~x TABLE

(Columns 1, 2, 3, and 4)12-inch Gun and Long Pointed Projectile

Height of Site, 200 feet '

Range ~lyds. deg. min. deg. min.

8000 5 15.2 31.4 43.8 4 41.820 16.2 31.3 44.9 42.840 17.1 31.2 45.9 43.960 18.1 31.2 46.9 44.980 19.0 31.1 47.9 45.9

8100 19.9 31.0 48.9 46.920 20.9 31.0 49.9 48.040 21.8 30.9 50.9 49.060 22.8 30.8 52.0 50.080 23.8 30.8 53.0 51.1

8200 24.7 30.7 54.0. 52.120 25.7 30.7 55.0 53.140 26.6 30.6 56.0 54.260 27.6 30.5 57.1 55.2

'80 28.530.5 58.1 56.28300 29.5 30.4 59.1 57.3

I

~

-



36 DIRECT FIRE

Range I() -I:i.SOhK SOx SOlyds. deg. min. min. deg. min. deg. min.

20 5 30.5 3004 5 00.1- 4 58.340 31.4 30.3 01.1 59.460 32.4 30.2 02.2 5 00.480 33.3 30.2 03.2 01.58400 34.3 30.1 04.2 02.520 35.3 30.1 05.2 03.540 36.3 30.0 06.3 04.660 37.2 29.9 07.3 05.680 38.2 29.9 08.4 06.78500 39.2 29.8 09.4 07.720 40.2 29.8 lOA 08.840 41.2 29.7 11.5 09.960 42.1 29.6 12.5 10.980 43.1 29.6 13.5 11.98600 44.1 29.5 14.6 13.020 45.1 29.5 .15.6 14.140 46.1 29.4

16.7 15.160 47.1 29.4 17.7 16.280 48.1 29.3 18.8 17.28700 49.1 29.3 19.8 18.320 50.0 29.2 20.8 19.340 51.0 29.2 21.9 20.4,60 52.0 29.1 22.9 21.580 53.0 29.0 24.0 22.58800 54.0 29.0 25.0 23.620 55.0 28.9 26.124.740 56.0 28.9 27.1 25.760 57.0 28.8 28.2 26.880 58.0 28.8 29.2 27.98900 59.0 28.7 30.3 28.920 6 00.0 28.7 31.3 30.040 01.0 28.6 32.4 31.160 02.0 28.6 33.4 32.280 03.0 • 28.5 34.5 33.39000 04.0 28.5 35.5 34.3

Assume the following: Range to target, 8500 yards;tide, 0; wind, none; atmosphere, normal; muzzle velocity,2250 f.s.; projectile, long pointed; gun, 12-inch.

. The piece should be given a quadrant elevation of SOl

5° 7'.7. (The range setting will be 8500 yards.) When thepiece is fired, the average jump will give a quadrant angle of

departure of SOx 5° 9'.4.Fig. 6 shows diagrammatically the relations between "the

range table angle of departure, so, and the quadrant angle of

-

=

=



DIRECT FIRE

~--~

f/J = li'ang'e hole J-blue.~' = N on -R an tye bb le /-b /ve

37

FIG. 4.

or 9? f/el/al/on h.6lf? J/alue.

rfJ/ or ~ '=N on.EleJ/of/o/1 lable I/o/vB.

FIG. 5.

I5n;)

1668

6'

_..J:K+ A sPh--- ----------FIG. 6.

FIG. 7.

T '

T

I61i8

1667

~ =



38 DIRECTFIRE

departure, CPx. A sharp distin~tion must be drawn betweenthe quadrant angle of elevation, ((;I, and CPx. It is illustratedin Fig. 7. AcpJ may be either plus or minus. As shown, thejump is positive, wherefore the correction for jump is -Al(/j,

in order that when the piece is fired the elevation shall in-crease from CPt to the proper quadrant angle of departure CPx,in the direction of jump.

RANGE CORRECTIONS

vVe are in a position to calculate CPx with sufficient accuracy.

for any gun, for any range, and for any conditions.See Figs. 4, 5, and 8.

If the target is in the initial plane and either V or Cchanges from range table values, we can still hit the targetby altering cp by the proper Acp, as illustrated in Problem 1 to7, inclusive (see Fig. 4).

It is assumed that by applying the same Acp (= Acpx) tocpz we can cause the shot to range to the same horizontal dis-tance under the same variations in V and C, no matter howgreat the height of site of the gun within service limits (seeFigs. 5 and 8).

The order of procedure can be stated to be as follows:1. For any non-range-table Condition of V and C, assume

the target to be in the initial plane, and solve for the required.p •

2. Take cp for the range from the range table by inter-polation where necessary. (N OTE: It is preferable for thispurpose not to use the value in .the column headed "Changein elevation for 10 yards in range.")

3. cp' cp Acp.

4. Determine CPx for the range from your computedR-cpz table for long-pointed projectiles (or CPE from the quad-rant elevation table for capped projectiles).

- =



DIRECT FIRE 39

5. To 'Px (or 'PE) algebraically add D..'P to get cp'x (or CP'E)'

6. In your computed R-cpx table for long pointed pro-jectiles (or 'PE table for capped projectiles), determine by

interpolation the range corresponding to your 'P'x (or CP'E)'When the gun is set for this range you will have the properquadrant angle of departure, or angle of departure.

7. From the range setting determined in 6, subtract the .actual range to the target. This will give the AR (range c,or-rection reference number) which, when applied to the rangescale, will cause the proper change in the ~ngle of departure .

. Problem 11Given: 12-inch gun and long pointed projectile; range to

target~ T', 8500 yards; C=I1.7945; V=2215 f.s.; height ofsite, 200 feet.

1. Solve for target as if in the initial plane ..

Z=XC

log X log 25500 4.40654

colog C 8.92832 -10log Z 3.33486

Z 2162Sin 2'P' AC

. For Z 2162. A A(V 2200) .01739 -78(V = 2250) .01661

15(V 2215) A .01739 -78 X50 =.01716.

log A log .01716 8.23451 -10log C 1.07168

log sin 2'P' 9.30619-102cp' 11° 40'.6

cp ' 5° 50'.3

2. Referring to the range table (Form 1014), we see thatcp 5° 39'.2 for 8500 yards.

3. Therefore, to compensate for' the 35 f.s. reduction invelocity, we must increase our angle of departure by 'P' 'P

5° 50'.3-5° 39'.2= +11'.1 =AIp.

4. Referring to our elevation table, we see that thequadrant angle of departure,lpf, for 8500 yards is 5°09'.4.The angle of departure must be increased by A Ip, wherefore,

5. 'P'x (Px+ Alp 5°09'.4 + 11'.1 = 5°20'.5.

= = =

= =

= =

=

= = = =

=

= = =

=

- =

= ~



40 DIRECT FIRE

6.. What range setting will give us this angle of departure~5°20'.5?

Referring again to the R-A x table we see by. interpo-lation that the range setting will be

8700 +20 X ;g:~=~~:~= 8700 +20 X 1:0 = 8714 yards.

7. Therefore the change in the angle of departure isobtained by a change in the range setting of

8714 8500 +214 AR on the range scale.This is the range correction reference number to .be

applied at the. range scale.In the operation of a range board fitted with a range cor-

rection chart for the 12-inch gun and long pointed projectile,it will be found that the correction to be applied on the rangescale will be very close to + 207 for a gun on a height of

site of 200 feet.

The latest range correction charts for capped projectilesare computed in a like manner, excepting that the A«J is ap-plied to «JEto determine the range setting.

For a given variation from range table conditions, the~R to be applied on the range scale will vary with the heightof site, because the relation between Rand «Jx,or «JE,varies asthe height of site varies.

In other words, the same elevation or R-«Jx tables willnot do for 12-inch guns and long pointed projectiles for all

heights of site; nor for 12-inch guns and capped projectiles forall heights of site; not for long pointed projectiles and cappedprojectiles for any height of site. And, because we apply~«J to «Jxor «JEto interpolate for the new range setting, thechange in range setting, ~R, will vary with variations in heightof site.

Finally, this AR .at the range scale is a reference numberwhich secures a proper change in «Jxor «JEby changing the

range setting. Adding 200 yards to the range setting maybe expected to increase the range of the shot by 200 yards onlyunder normal conditions. Where the V or C varies fronlnormal, the range correction on the range board will give forany range the range correction reference number which, whenapplied at the range scale, will produce the proper change inthe angle of departure.

Problem 12Given: 12-inch gun and long-pointed projectile; range to

- = =



DIRECT FIRE41

AR =8444 -8500 -56

target 8500 yards; V, 2250; C, 11.7945; height of site, 200 feet;atmosphere reference number, 20.

What is the required range correction reference number?An atmosphere reference number of 16 represents a

normal atmosphere. Under the conditions, there is an in 4

crease of 20 -16 =4% in the ballistic coefficient due to atmos- .pheric conditions as determined by the barometer, thermom-eter,and atmospheric slide rule.

log C 1.07168log 104% 0.01703

log C 1 1.08871, for the day, theactual value we have to work with under the conditions.

Z=XC 1

log X 4.10654colog C 1 8.91129 -10

log Z 3.31783Z 2079

Sin 2q,' AC

(V =2250; Z =2079) A =.01513+ 91 X Z~O=.01585

log A 8.20003 -10log C] 1.08871

log sin 2q/ 9.28874 -102cp' 11° 12'.7

cp' 5° 36'.3

2. From the range table, for 8500 yards, cp =5° 39'.2.3. Acp=cp'-cp=5° 36'.3-5° 39'.2= -2'.9.4. Referring to our R-cpx table, we see that the quadrant

angle of departure, CPx,or 8500 yards is 5° 09'.1.5. cp'x CPx+Acp=5° 09'.1+( -2'.9) =5° 06'.5.6. Referring to our R-cpx table, we see that a quadrant

angle of departure of 5° 06'.5 corresponds to a range setting of

8440 _ 06.5 -06.3 X20 =844407.3 06.3 ...

7. Therefore the range correction reference numberwhich will secure the proper change in the ql1adrant angle ofdeparture is

=

= =

=

= =

= =

=

= =

= = =

=

-



42 DIRECT FIRE

log X = 4.40654colog C 1 8.95519 -10

log Z 3.36173Z = 2300

Problem 13

Given: 12-inch gun and long pointed projectile; range totarget, 8500 yards; C, 11.7945; V, 2250; wind range componentreference number, 20; height of site, 200 feet.

What is the required range correction reference .number?A range component reference number of 50 denotes that

either the direction of the wind with reference to the line offire is 90°, or that the wind has 0 velocity; in either case thereis little or no movement of the atmosphere parallel to the linejoining gun and target.

Therefore 20 50 = 30 miles per hour velocity of windagainst the motion of the projectile.

The effect of the wind is to decrease the value of theballistic coefficient.

/. 1::l:: 2W xT5/4. . X

(log T log 13.11 1.11760)5/4 log T 1.39700

log 2W x log 60 1.77815colog X = 5.59346 -10

log 0.0587 8.76861-10Since it is a head wind,f.. 1 .059 .9413 .94

log C 1;07168log 0.94 9.97313 -10

log C 1 1.04481

Sin 2lp' AC(V 2250; Z = 2300) A .01790

log A = 8.25285 -10log C 1 1.04481

log sin 2lp' 9.29766 -102lp' 11° 26'.8q/ = 5° 43'.4

=

=

-

=

= = =

= =

=

= - = = = =

=

= = =

=

= =



DIRECT FIRE 43

1:11()= I() ' -I() =5 0 43'.4 _50 39'.2 =4'.2I()'x I()"+dl() 50 09'.4 +4'.2 = 50 13'.6

The range setting corresponding to an angle of departureof 5° 13',6 is, from our R-lpx table, 8580.

Therefore the dR to be applied to the range scale to pro-duce the proper change in the angle of departure is

AR 8580 8500 + 80

Problem 14

Given: a 12-inch gun and long pointed projectile; range

to target, 8500 yards; C~ 11.7945; V, 2250'; tide, + 10 feet;height of site, 200 feet.

What is the required range correction reference number?Since the basic condition is mean low water, we must

determine ho~ much to increase the angle of departure so asto raise the point struck 10 feet.

o 10Tan AI()=y

log 10 1.00000colog X = 5.59346 -10

log tan AI() 6.59346 -10AI() 1'.4

lp'x lpx+AI() =5 0 09'.4+1'.4 =5 0 10'.8The range setting giving an angle of departure of 50 10'.8

is

+40

90

NV+200

324

70

+20 ' +30

50

16

+10

8520 +20 X i~ 8527

The range correction reference number is, therefore,

8527 8500 =+ 27Problems 11 to 14 inclusive illustrate how range correction

refe,rence numbers are computed for a particular height of .site. For a range correction chart the computations are madefor every 1000 yards within the limits of quadrant elevationpermitted by the carriage, and for each of the following vari-ations from range table conditions:

(a) Denote the normal velocity by NV:NV-200 NV-100 NV NV+100

(b) For atmosphere:o 8

(c) For wind:10 30

(d) For tide:-10

= =

= - =

=

= =

=

=

-



44 DIRECT FIRE

The straight vertical. lines representing the ~ormal C9n-ditions are drawn at convenient distances apart, so that thecurves shall not cross. The horizontal range lines are drawnfor every 1000 yards, and are spaced to a vertical scale of1 inch 600 yards. The horizontal scale of the latest chartsis 1 inch 200 yards.

The reference number for any abnormal condition isplotted on the horizontal range line to a scale of 1-inch 200,

from the proper normal line as zero. The reference numbershaving a + value to the left of the normal; those havingvalue to the right of the normal: Curves are then drawnthrough all points representing the corrections' at every 1000yards.' For example, having plotted in each range line thecorrections for a 70 wind, all such points are connected by asmooth curve. The curve for the 60 wind is plotted anddrawn half way between the 50 (normal) and the 70 curve.

For issue to the service it has been determined that onechart for a given gun, projectile, and normal muzzle velocityis sufficiently accurate between each of the following limits ofheight of site: 0 to 40 feet; 40 to 60 feet; 60 to 100 'feet; 100to 160 feet; 160 to 240 feet; 240 to 320 feet; 320 to 400 feet;400 to 500 feet; 500 to 600 feet. For each gun, projectile,and normal muzzle velocity .there are nine charts. The par-ticular chart for the battery depends on the limits of heightof site between which it lies. The chart for batteries (havinga given caliber, projectile, and normal muzzle velocity) onheights of site between 40 and 60 feet are computed for aheight of site of 50 feet, or the meari height. Similarly forthe others.

I t will be seen that the range correction reference numberdoes not correspond to the horizontal range correction at the.target, when either

V,C, or tide is abnormal. To illustrate:

From Problem 11, we see that an increase in the angle ofdeparture of A lfJ + 11'.1 is required for a muzzle velocityof 2215 f.s., or a decrease from normal of 35 f.s. We havedetermined that the range correction. to .be applied to therange scale to produce an increase of 11'.1 in the angle ofdeparture is + 214.

We see from the range table that for 8500 yards, lfJ 5°

39'.2.lfJ' ,.;, 5° 39'.2 + 11'.1 5° 50'.3

The range in the range table corresponding to 5° 50'.3 is8725.

= =

=

-

=

=

=



DIRECT FIRE 45

8725 8500 + 225

Or a change in the range setting of +214 will change thehorizontal range of the shot by +225 yards. .

SOLUTIONS FOR M UZZLE VELOCITY

Given: a 12-inch gun and long pointed projectile; rangeto the aiming point, 8500 yards; range table value of C,11.7945; atmosphere reference number, 20; wind referencenumber,' 20; tide, +10; range to the center of impact, 8200

yards; height of site, 200 feet.What was the muzzie velocity?

SOLUTION I

Find the angle of departure.The angle of departure is resolved into two parts: (1) the

quadrant angle of departure, 'P' x, or that part above thehori-zontal plane through the trunnions; and (2) the angle of de-pression corresponding to the range to the center of impact,or that portion below the horizontal plane.

Referring to problems 11 to 14, we see that the net rangecorrection reference number given by the range board is

214--56 +80+27 + (2000) +265 (2265)_To suit our condition to mean low tide, for which our

elevation and R-'Px tables are computed, we can consider ourrange correction reference number to be 265 27 (the tide cor-

rection) -238, and can consider that the range to the centerof impact would have been 8200 at mean low water, had therange correction reference number been 2238 instead of 2265.

The corrected range becomes 8500 +238 =8738.. 'P' x corresponding to a range setting of 8738 is 50 21 '.8.

Also, from the R-'Px table, we see that the depressionangle for the range to the center of impact is 30'.7.

Therefore,'P' =50 21 '.8 +30'.7 = 50 52'.5 .

. From problems 12 and 13,

C t fw X~X C 1.04 X 0.94 X Colog C 1.07168

log 1.04 0.01703log 0.94 9.97313 -10

log C 1 1.06184

- =

=

-

=

= =

= = =

=



---46

,.,DIRECT FIRE

.XZ C

1

log X log (3 X 8200) 4.39093colog C t 8.93816 -10

log Z 3.32909Z 2133

A_ sin 2 tp'

C1

log sin 2 tp' log sin 11 0 45'.0 9.30886 -10colog C 1 = 8.93816 -10

log A 8.24702 -10A 0.01766

We now enter Table II and determine the consecutivevelocities for which, for our value of Z 2133, the values ofA bracket our value of A. Then we interpolate for thevelocity by means of A.

Z 2133 (V 2200) A 0.01712 A 70(V 2150) A 0.01782

V 2200 50 X .01706-.01712 = 2200-39 = 2161 f.s.. .00070

NOTE: If a range correction chart is at hand, the per-centage change in C effected by the wind can be determinedquickly, as follows, without use of the wind formula: Holdthe straight edge of a piece of paper parallel to the range linesand at the range of the aiming point. Graduate the edgewhere it is intersected by the normal (50) and letter the markO. Gradp.ate the edge where the wind curve (or interpolatedimaginary curve) cuts the edge. Letter the graduation K.

Transfer the paper parallel to the range lines until the0

graduation is over the normal atmosphere curve (16). TheK graduation n'ow indkates the equivalent change in C interms of an atmosphere reference number, whence the per-centage change is determined by remembering that 16 is1.00, 18 is 1.02, 14 is .98, etc .

.If the chart is mounted on a range board, the operationmay be performed with the ruler.

SOLUTION II

In this case, consider that the change in the ballisticcoefficient due to atmosphere and wind is compensated forby their portions of the total range correction reference num-

= = =

=

= =

-

= =

= =

=

= = = = = =

= -

-





48 DIR EC T FIRE

1. Select the aiming point for the trial shots and deter-

mine its range. .2. Determine the muzzle velocity to be assumed for trial

shots.3. Assume centers of impact at 100 yard intervals over

and short, within limits of about ::l= 500 yards.4. Compute and tabulate the velocities corresponding to

such rangings.5. On cross-section paper, with the X axis for range and

the Y axis for velocities, plot these points and connect them

by a smooth curve.Deflections

The deflection problem is briefly as follows (see Fig. 9):Assume a target at T moving at such a rate in the direction

shown by the arrow that it will be at T 1 at the end of the timeof flight for the range GT. It is obvious that we must inclinethe plane of departure to the plane of sight by the angle 1 tocorrect for the angular movement of the target during tl:e

time of flight.Assume that, due to drift alone, at this range, the point

of impact would fall in the direction of T 2 instead of T. Itis obvious that we must increase the angle 1 by the angle 2

to correct for drift also.Assume that the wind would blow the projectile so as to

cause the point of impact to be in the direction of Tg insteadof T. It is obvious that we must increase angle 1 by theangle 3 to correct for the effect of the wind as well.

Therefore, the net angular correction for travel duringtime of flight, drift, and the lateral wind component is the

angle 4 (= 1 + 2 + 3). .Then, if the plane of departure is inclined to the plane of

sight by the angle 4 as set off on the sight, the projectile willdepart from the plane G T 4 under the action of wind and driftand will strike in the direction of the point T h the directionof the target at the end of the time .of flight. .

For simplicity, the figure is drawn showing the conditions

to require cumulative corrections.In Cases I and II the required divergence of the plane of

departure from the plane of sight is secured by means of the

deflection set off on the sight.. In Case III, the angle 4 is applied to the uncorrecte?

azimuth of the line G T in order to obtain the corrected aZI-muth GT

4at which to lay the piece. This applies to both



DIRECT FIRE 49

guns and mortars. A study of the deflection board will

illustrate how these corrections are automatically added.On a gun deflection board note that:(a) The drift leaf is only for certain projectiles, caliber,

and muzzle velocity.(b) The scale on the T-square is not equicrescent and the

ranges are graduated in proportion to their time of flight, andtherefore the'T-square is only for certain projectiles, caliber,and muzzle velocity.

Rules for deflection corrections:R-R. Right-Raise. To shoot more to the Right, Raisethe value of the deflection or corrected azimuth by the desired

amount.If the shots are falling 0°.10 to the left of the target,

Raise the value of the deflection or corrected azimuth 0°.10in order' to shoot 0°.10 more to the Right.

FIG. 9.1560

L-L. Left-Lower. To shoot more to the Left, Lowerthe value of the deflection or corrected azi,muth by the desired

amount.If the shots are falling 0°.15 to the right of the target,

Lower the value of the deflection or corrected .azimuth 0°.15in order to shoot 0°.15 more to the Left.

To avoid confusion and errors in applying corrections, thecorrected deflection, not the correction, should be transmitted.

The smallest division on the sight is 0.°05. The naturaltangent of 0°.05 (=3') is approximately 1/1000, but the truevalue to sufficient accuracy is .00087.

The latter value should be used for cOIp.puting deviationsin yards from observed angular deviations.

The approximate value of .001 is of interest in makingquick computations. 1 division 1 yard per thousand ofrange.

=



50DIRECT FIRE

ExampleA battleship is head-on at 8000 yards. The shots seem

to fall consistently about half the width of the target to the

right. Deflection: 2.80.Solution:One width from the center of the target equals about 90

feet 30 yards.One division on sight at 8000 yards 8000 X .001 8

yds.30 + 8 = 4 aivisions.Correction 4 X ..05 0°.20To shoot Left-Lower. Corrected deflection 2.80

.20 = 2.60DIFFERENTIAL FORMULlE

Occasionally it is desirable to know how much of a range

error is to be expected from a variation from normal, eitherin the weight of the projectile, or of the charge, or both. Thefollowing formulre are fairly accurate for small variations fromstandard conditions. .

Variation in muzzle velocity due to a variation in weight of

project; Ie:(1) A V 1- Aw V A V _.l- Aw V I

16 w 4 w

Problem 16An officer is to fire long pointed projectiles from his 12-

inch battery for the first time. Previous firings with thesame weight of powder and capped (1046-lb.) projectilesindicate that for the temperature of 70° F., the M.V. assumedshould be 2220 f.s. . About what M.V. will be given to thelong pointed projectiles weighing 1070 lbs.?

7 24A V 16 X 1046 X 2220 22 f.s.

The result indicates that he should assume about 2200 f.s.

Problem 17A battery commander is to fire long pointed .projectiles

from his 12-inch battery for the first time. Previous firingswith the weights of powder as made up into charges, and using

eapped (1046-lb.). projectiles, indicate that at a temperatureof 70°F. the M.V. to be assumed is 2220 f.s. Each charge asmade up weighs 276.5 Ibs. About what weight of nitro-cel-lulose powder should be added to each charge to give the nor-

mal M.V.?

= = =

= = = -

= - =

= - = -



DIRECT FIRE 51

(2)

From Problem 16 the M.V. will probably be approxi-mately 2200 f.s.at 70° F.

The velocity to be expected due to changes in the weightof powder charges may be computed from the formula,

VV1

(WW1

)Y (See C.A.D.R., 1909, par 796.)

The average value of y is: for nitro-cellulose powder, 1.~;

for nitro-glycerin powder, 0.8.Therefore, 220q (276.5) 1.2

2250 WI

2250 X 276.5 1•2

Wt1

•2 2200

log 2250 3.35218. 1.2 log 276~5 2.93003

colog 2200 6.65758 -10

1.2 log WI 2.93979log WI 2.44982

WI 281.7Ibs. what each chargeshould weigh.

281.7 276.5 5.2 pounds to be added to each charge.Variation in muzzle velocity due to a variation in the travel

of the shot while in the bore:

(3) tl V =.l- tluV V= 41 tlUUV

16 uIn which u travel of shot. ~u variation in travel

of shot.Variation in muzzle velocity due to variation in size oj

chamber:(4)~ v= ~tlv V V=J..- tlVV

4 v 3 vIn which v volume of chamber. ~v variation in

volume of chamber.

Variation in muzzle velocity due to a variation in caliber:l5) V .l ~d V

8 d

In which d caliber. tld variation in caliber.In American guns the weight of projectile and charge

should be taken in pounds, the length of travel and caliber ininches, and the volume of chamber in cubic inches. Equa-

tion 5 is not very reliable.The coefficients used in these formul::e, were obtained for

=

=

= = = =

= = = =

- =

~

= =

~

= =

~ =

= =



52 DIRECT FIRE

(1)

(3)

(4)

nitrocellulose powder, and for small variations may be consid-

ered as approximately true for smokeless powders. EquationsAV=_l-AW V

4 w

A V= AuU V

A V=l- AU V3 U

are furnished by Major Alston Hamilton. C. A. C.

CHANGE OF HEIGHT OF SITE DUE TO SWEEP

OF THE MUZZLE .

Theoretically, the trajectory starts at the muzzle. As wechange the inclination of the bore, it will be observed thattheoretically we may consider the height of site to be changed.Practically, this has no bearing. It will be observed from the"method of solving for a range table C, as illustrated by Problem

8, that such sweep of the muzzle is neglected. Thus, anyerror introduced is compensated in the. determination of i.This conception is further strengthened by the fact that thebatteries at the proving ground are on comparatively low sites.To be consistent with the methods involved in securing datafor our range tables, we are therefore correct in considering theheight of site of a piece as that of the axis of the trunnions.

This point is mentioned because of the speculation whichthe idea has aroused in the past.

The following is quoted from Coast Artillery SchoolInformation Series No.1, Sept. 1, 1914:- .

I. CORRECTION FOR GUN DISPLACEMENT AND CARRIAGE OUT OF LEVEL

The following method of making thi's correction was devised by thebattery commanders at Fort Monroe, and some of the members of the class.

1. Select a point in the field of fire of the gun at about mid-rangeand for which the correction for gun displacement is zero.

2. Set the gun to the azimuth of this point, using the nearest fulldegree. This may be called the reference azimuth.

3. By the clinometer bring the gun to level and set the index properly.4. Traverse the gun right. and left, between the stops, halting at

every 10 degrees, and determine the amount, plus or minus, that the car-riage is out of level. Tabulate the above readings for each azimuth.

5. On a piece of cross-section paper layoff along the X-axis a scaleof degrees 01" ==10degrees), and layoff on the Y.axis a scale of yards(1" =20 yards) from minus 100 to plus 100. '. I'm

6. Opposite each 10-degree division plot the correction in yardsnecessary to compensate for the amount that the carriage is out of level,assuming a target at the range of the selected point. Connect these pointsby a smooth curve. Letter this curve "Level Correction."

~

~



DIRECT FIRE 53

7.• Opposite each 10-degree division plot the correction in yards nec-. essary to compensate for gun displacement. Connect these points by a

smooth curve. Letter this curve "Displacement Correction."8. Now draw the net correction curve by combining the level and.

displacement corrections.It was found that at Battery Eustis the net correction curve is sensi-

bly parallel to the X-axis, which permitted a flat correction to be appliedby shifting the elevation pointer.

I t will be noted that where a flat correction cannot beapplied with sufficient accuracy, the pointer can be shifted tocompensate for the corrections obtaining in the most importantpart of the field of fire. On the basis of this shift, the resultantcorrections for other azimuths can be determined and letteredas heretofore, thus reducing the amount of the correction tobe applied at the range scale. In some rases it may be deemedadvisable so to shift the pointer that all corrections shall be ofa positive value, thus eliminating the necessity of applying nowa positive correction, and at another time a negative correc-tion.

II. CORRECTION TO ADJUST THE CENTER OF IMPACT TO THE

CENTER OF THE RECORD TARGET FOR MAJOR CALIBER GUNS

1. The danger space in yards 13 cot (w+e)D.S.

2. The number of yards to be added -2--5

(Note: These formulae have been changed from previous editions to apply to the presenttargets prescribed for Coast Artillery target practice.)

Plot these values as determined for every 2000 yards from 3000 to10,000 yards; scale: X-axis, 1" 1000 yards; Y-axis, 1" 20 yards. Con-nect these points by a smooth curve. This chart was found convenientin plotting the target for the determination of hits.

These corrections vary with the range. They are incorrect for applica-tion as a range correction until the !lip corresponding thereto is determinedfrom the range table. With this !lip as an argument, the !lR to be appliedat the range scale is determined from the R-<px, or R-IpE, table for the bat-tery.

3. The correction expressed in feet of tide is,!lR t 3 X correction in yards X tan(w +e)

4. To adjust the center of impact to the center of the record target, achart was made showing the above corrections expressed in feet of tide, byplotting the corrections for every 3000 yards from zero to 11,000 yards andconnecting the points by a smooth curve.

It was seen that a flat correction could be made by~the tide curve fortarget practice ranges without material error. For Battery Eustis thecorrection was about 13 feet of tide.

I t will be observed that the greater tide curves on thelatest range correction charts were plotted up to + 40 feet of

tide for this purpose, and to meet the service condition inplacing the center of impact at the desired point on a hostiletarget.

=

=

= =

=



54 DIRECT FIRE

The 1914 record target is described in the Regulations

for the Instruction and Target Practice of Coast Artillery Troops, .1914 as follows:123. Record Target for Alajor and Intermediate Caliber Guns.-Using

a piece of cross-section paper at a scale of 50 yards to the inch, considerone of the heavy lines as the track of the target and the intersectionof the two heavy lines as the location of the towed target. On thetrack of the target layoff the length of the towline and plot the positionof the tug. Draw a line of direction through the center of the target andthe directing point of the battery. Draw two lines parallel to and 15 yardsfrom the track of the target. Similarly, draw two lines parallel to and 15yards from the line of direction. The parallelogram so formed is the hor-izontal projection of a section of a battleship, and is part of the recordtarget. To complete the record target extend the sides of this parallelo-gram 10 yards toward the battery and the length of the danger spaceaway from the battery, assuming the freeboard of a battleship as 30 feet.Thi~ completed parallelogram is the record target.

TA B L E A

Values of aI/a for temperature and pressure of atmosphere78% saturated with moisture. (From Artillery Note No. 25.)

her. BarometerF. 28 11 29" 30" 31"

5° 0.942 0.909 0.880 0.8516° 0.944 0.911 0.881 0.853

7° 0.946 0.913 0.883 0.8558° 0.948 0.915 0.885 0.8569° 0.950 0.917 0.887 0.858

10° 0.952 0.919 0.889 0.86011° 0.954 0.921 0.890 0.86212° 0.956 0.923 0.892 0.86413° 0.958 0.925 0.894 0.86614° 0.960 0.927 0.897 0.86715° 0.962 0.929 0.899 0.86916° 0.964 0.931 0.901 0.87117° 0.966 0.933 0.903 0.87318° 0.968 0.935 0.905 0.87519° 0.971 0.937 0.907 0.87720° 0.973 0.939 0.909 0.87921° 0.975 0.941 .0.911 0.88122° 0.977 0.943 0.912 0.88323° 0.979 0.945 0.914 0.88524° 0.981 0.947 0.916 0.88725° 0.983 0.949 0.918 0.888

26° 0.985 0.951 0.920 0.89027° 0.987 0.953 .0.922 0.89228° 0.990 0.955 0.924 0.89429° 0.992 0.958 0.926 0.896

Barometer T28" 29" 30" 31"

0.890 0.861 0.831 0.8060.892 0.863 0.833 0.8080.894 0.864 0.835 0.8090.896 0.866 0.837 0.8110.898 0.868 0.839 0.8130.901 0.870 0.841 0.8150.903 0.872 0.843 0.8160.905 0.874 0.845 0.8180.907 0.876 0.847 0.8200.910 0.878 0.848 0.8220.912 0.880 0.850 0.8240.914 0.881 0.852 . 0.8260.916 0.883 0.854 0.8270.918 0.885 0.856 0.8290.920 0.887 0.858 0.8310.922 0.889 0.860 0.8330.924 0.891 0.862 0.8350.926 0.893 0.864 0.836

0 0.928 0.895 0.866 0.8380.930 0.897 0.868 0.8400.932 0.899 0.870 0.842

0.934 0.901 0.871 0.8440.936 0.903 0.873 0.8450.938 0.905 0.876 0.8470.940 0.907 0.878 0.849

Ther.F.

-20°-19°-18°-17°-16

-1 5-14

-1 3-12

-1 1-10

-9

-8

- 7

- 6

- 5

- 4

- 3

- 2

-1o1234

° ° ° ° ° ° ° °

° ° ° ° ° °

° °

° ° ° °



DIRECT FIRE

TABLE A-Continued.

55

Ther. Barometer Ther. BarometerF. 28" 29" 30" 31" F. 28" 29" 30" 31"

30° 0.994 0.960 0.928 0.898 66° 1.073 1.035 1.001 0.96831° 0.996 0.962 0.930 0.899 67° 1.075 1.037 1.003 0.97032° 0.998 0.964 0.932 0.902 68° 1.078 1.040 1.005 '0.97333° 1.000 0.966 0.934 0.903 69° 1".080 1.042 1.007 0.97534° 1.003 0.968 0.936 0.906 70° 1.082 1.044 1.009 0.977.35° 1.005 0.970 0.938 0.907 71° 1.085 1.046 1.011 0.97936° 1.007 0.972 0.940 0.909 72° 1.087 1.048 1.013 0.98137° 1.009 0.974 0.943 0.911 73° 1.089 1.050 1.015 0.98338° 1.011 0.976 0.945 0.913 74° 1.092 1.053 1.017 0.98539° 1.013 0.978 0.947 0.915 75° 1.094 1.055 1.019 0.98740° 1.015 0.980 0.949 0.917 76° 1.096 1.057 1.022 0.98941° 1.017 0.982 0.951 0.919 77° 1.099 1.059 1.025 0.99242° 1.019 0.984 0.953 0.921 78° 1.101 1.062 1.027 0.99443° 1.021 0.987 0.955 0.923 79° 1.104 1.064 1.029 0.99644° 1.023 0.989 0.957 0.925 80° 1.106 1.066 1.031 0.99845°

1.026 0.991 0.959 0.927 81° 1.109 1.068 1.033 1.00046° 1.028 0.993 0.961 0.929 82° 1.111 1.071 1.035 1.00247° 1.030 0.995 0.963 0.931 83° 1.114 1.074 1.038 1.00548° 1.033 0.997 0.964 0.933 84° 1.116 1.076 1.041 1.00749° 1.035 0.999 0.966 0.935 85° 1.119 1.079 1.043 1.00950° 1.037 1.002 0.968 0.937 86° 1.121 1.081 1.045 1.01151° 1.040 1.004 0.970 0.939 87° 1.124 1083 1.047 1.013

'52° 1.042 1.006 0.972 0.941 88° 1.126 1.086 1.049 1.01653° 1.044 1.008 0.974 0.943 89° 1.129 1.089 1.053 1.01854° 1.046 1.010 0.976 0.945 90° 1.131 1.092 1.055 1.02055° 1.048 1.012 0.978 0.947 91° 1.134 1.094 1.057 1.02256° 1.050 1.014 0.980 0.949 92° 1.136 1.096 1.059 1.02557° 1.053 1.016 0.982 0.951 93° 1.139 1.099 1.062 1.02758° 1.055 1.018 0.984 0.952 94° 1.142 1.102 1.064 1.02959° 1.057 1.020 0.986 0.954 95° 1.144 1.105 1.066 1.03160° 1.059 1.022 0.988 0.956 96° 1.147 1.107 1.068 1.03361° 1.062 1.025 0.990 0.958 97° 1.149 1.110 1.071 1.03562° 1.064 1.027 0.992 0.960 98° 1.152 1.112 1.074 1.03763° 1.066 1.029 0.994 0.962 99° 1.155 1.115 1.076 1.040

64° 1.068 1.031 0.996 0.964 100° 1.157 1.117 1.079 1.04265° 1.071 1.033 . 0.998 0.966







58 DIRECT FIRE

TABLE B

(Taken from Ingalls' Table II, Artillery Circular M)

V 2100 f.s.

3

445

434

Xlog B' .1 z .1 z T' .1 z log C' .1 z

z= - uC

-10100 0.0037 37 2073 26 0.048 49 4.8663 37200 0.0074 37 2047 26 0.097 49 4.8700 37300 0.0111 37 2021 26 0.146 50 4.8737 37

..400 0.0148' 37 1995 25 0.196 50 4.8774 38500 0.0185 37 1970 25 0.246 51 4.8812 38600 0.0222 37 1945 25 0.297 52 4.8850 38

700 0.0259 37 1920 25 0.349 52 4.8888 38800 0.0296 37 1895 24 0.401 53 4.8926 39900 0.0333 38 1871 24 0.454 54 4.8965 39

1000 0.0371 38 1847 24 0.508 54 4.9004 391100 0.040938 1823 23 0.562 55 4.9043 391200 0.0447 38 1800 24 0.617 56 4.9082 , 40

1300 0.0485 38 1776 23 0.673 57 4.9122 391400 0.0523 38 1753 23 0.730 57 4.9161 401500 0.0561 3~ 1730 22 0.787 58 4.9201 40

1600 0.0600 38 1708 22 0.845 59 4.9241 411700 0.0638 38 1686 22 0.904 60 4.9282 401800 0.0676 38 166422 0.964 61 4.9322 41

1900 0.0714 38 1642 22 1.025 61 4.9363 412000 0.0752 38 1620 21 1.086 62 4.9404 412100 0.0790 39 1599 21 1.148 63 4.9445 41

2200 0.0829 38 1578 21 1.211 64 4.9486 422300 0.0867 38 .1557 20 1.275 64 4.9528 422400 0.0905 38 1537 20 1'.339 65 4.9570 42

2500 0.0943 38 1517 20 1.404 67 4.9612 422600 0.0981 38 1497 20 1.471 67 4.9654 422700 0.1019 38 1477 19 1.538 69 4.9696 432800 0.1057 37 1458 19 1.607 69 4.9739 432900 0.1094 38 1439 19 1.676 70 4.9782 43000 0.1132 39 1420 19 1.746 71 4.9825 4

3100 0.1171 39 1401 18 1.817 72 4.9868 43200 0.1210 39 1383 18 1.889 73 4.9912 43300 0.1249 38 1365 18 1.962 73 4.9955 4

3400 0.1287 37 1347 17 2.035 75 4.9999 43500 0.1324 37 1330 17 2.110 76 5.0043 43600 0.1361 36 1313 16 2.186 76 5.0087 4

=

'

'

'



DIRECT FIRE

TABLE B


V 21D0 f.s.

59

Xlog B' Az Az T' Jog C' Az=-- u Az

C

-103700 0.1397 35 1297 16 2.262 78 5.0132 443800 0.1432 35 1281 15 2.340 78 5.0176 453900 0.1467 35 1266 15 2.418 80 5.0221 45

4000 0.1502 34 1251 15 2.498 81 5.0266 454100 0.1536 34 1236 14 2.579 81 5.0311 454200 0.1570 34 1222 13 2.660 82 5.0356 45

4300 0.1604 34 1209 13 2.742 83 5.0401 454400 0.1638 33 1196 13 2.825 84 5.0446 454500 0.1671 32 1183 12 2.909 85 5.0491 45

4600 0.1703 30 1171 .12 2.994 86 5.0536 . 454700 0.1733 29 1159 12 3.080 87 5.0581 454800 0.1762 28 1147 11 3.167 87 5.0626 46

4900 0.1790 28 1136 10 3.254 89 5.0672 455000 0.1818 26 1126 10 3.343 89 5.0717 465100 0.1844 24 1116 10 3.432 91 5.0763 45

5200 0.1868 24 1106 9 3.523 91 5.0808 455300 0.1892 24 1097 9 3.614 92 5.0853 45

5400 0.1916 23 1088 9 3.706 92 5.0898 45

5500 0.1939 22 1079 8 3.798 93 5.0943 445600 0.1961 21 1071 9 3.891 94 5.0987 445700 0.1982 20 1062 8 3.985 94 5.1031 44

5800 0.2002 20 1054 8 4.079 95 5.1075 445900 0.2022 19 1046 7 4.174 96 5.1119 436000 0.2041 17 1039 7 4.270 97 5.1162 43

6100 0.2058 16 1032 7 4.367 97 5.1205 436200 0.2074 16 1025 7 4.464 98 5.1248 436300 0.2090 15 1018 6 4.562 99 5.1291 42

6400 0.2105 15 1012 7 4.661 99 5.1333 426500 0.2120 14 1005 6 4.760 100 5.1375 426600 0.2134 14 999 7 4.860 100 5.1417 42

6700 0.2148 14 992 6 4.960 101 5.1459 41

6800 0.2162 13 986 6 5.061 102 5.1500 416900 0.2175 12 980 6 5.163 102 5.1541 41

7000 0.2187 11 974 6 5.265 103 5.1582 40

I

=

-- --- -- ---

~



60 DIRECT FIRE

TABLE B

(Taken fromlngalls' Table II, Artillery Circular M)

V 2200 f.s.

444

3

X log B' T' .1 z log C' .1 zz=- .1 z u .1 z

-10100 0.0036 36 2173 27 0.046 46 4.8266 35200 0.0072 36 2146 26 0.092 47 4.8301 36300 0.0108 37 . 2120 27 0.139 47 4.8337 36

400 . 0.0145 36 2093 26 0.186 48 4.8373 36500 0.0181 37 2067 26 0.234 49 4.8409 37600 0.0218 37 2041 26 0.283 49 4.8446 37

700 0.0255 37 2015 . 25 0.332 50 4.8483 37800 0.0292 37 1990 26 0.382 51 4.8520 37900 0.0329 37 1964 25 0.433 51 4.8557 38

1000 0.0366 38 . 1939 25 0.484 52 4.8595 381100 0.0404. 37 1914 25 0.536 52 4.8633 391200 0.0441 38 1889 24 0.588 54 4.8672 39

1300 0.0479 37 1865 25. 0.642 54 4.8711 391400 0.0516 38 1840 24 0.696 54 4.8750 391500 0.0554 38 1816 23 0.750 56 4.8789 39

1600 0.0592 38 .1793 24 0.806 56 4.8828 401700 0.0630 38 1769 23 0.862 57 4.8868 401800 0.0668 38 1746 23 0.919 58 4.8908 40. 1900 . 0.0706 39 1723 22 0.977 58 4.8948 412000 0.0745 38 1701 22 1.035 59 4.8989 412100 0.0783 . 38 1679 22 1.094 60 4.9030 41

2200 0.0821 39 1657 21 1.154 61 4.9071 412300 0.0860 38 1636 22 1.215 62 4.9112 422400 0.0898 38 1614 21 1.277 62 4.9154 412500 0.0936 38 1593

21 1.339 64 4.9195 42600 0.0974 38 1572 20 1.403 64 4.9237 42700 0.1012 38 1552 21 1.467 65 4.9279 432800 0.1050 I 38 1531 20 1.532 65 4.9322 42900 0.1088 38 151l 20 1.597 67 4.9364 43000 0.1l26 38 1491 20 1.664 67 4.9407 433100 0.1l64 38 1471 19 1.731 68 4.9450 43200 0.1202 38 1452 19 1.799

69 4.9493 43300 0.1240 37 1433 19 1.868 70 4.9536 4

3400 0.1277 38 1414 19 1.938 71 4.9579 43500 _. 0.1315 38 1395 18 2.009 73 4.9623 43600 0.1353 37 1377 17 2.0R2 73 4.9667 4

=



DIRECT FIRE 61

TABLE B

(Taken from Ingalls' Table II. Artillery Circular M)

V 2200 f.s.

Az log C' Az

-1074 4.9711 4475 4.9755 4576 4.9800 45

77 4.9845 4578 4.9890 4578 4.9935 46

80 4.9981 4581 5.0026 4681 5.0072 46

83 5.0118 4583

5.0163 4685 5.0209 46

85 5.0255 4686 5.0301 4687 5.0347 46

88 5.0393 4689 5.0439 4689 5.0485 46

90 5.0531 4691 5.0577 4592 5.0622 45

93 5.0667 4693 5.0713 4594 5.0758 45

95 5.0803 4596 5.0848 4596 5.0893 44

97 5.0937 4497 50981 4498 5.1025 44

99 5.1069 4399 5.1112 43

100 5.1155 43

101 5.1198 43

I

T'

2.1552.2292.304

2.3802.4572.535

2.6132.6932.774

2.8552.9383.021

3.1063.1913.277

3.3643.4523.541

3.6303.720 .3.811

3.9033.9964.089

4.1834.2784.374

4.4704.5674.664

4.7624.861

4.960

5.060

X10gB'=- Az u AzC

--3700 0.1390 38 1360 183800 0.1428 37 1342 173900 0.1465 38 1325 16

4000 0.1503 37 1309 164100 0.1540 37 1293 164200 0.1577 36 1277 16

4300 0.1613 35 1261 154400 0.1648 34 .1246 144500 0.1682 34 1232 14

4600 0.1716 33 1218 144700 0.1749 33 1204 134800 0.1782 32 1191 12

4900 0.1814 31 1179 125000 . 0.1845 31 1167 125100 0.1876 29 1155 11

5200 0.1905 29 1144 105300 0.1934. 27 1134 115400 0.1961 26 1123 10

5500 0.1987 26 1113 105600 0.2013 24 1103 95700 0.2037 23 1094 9

5800 0.2060 23 1085 95900 0.2083 21 1076 86000 0.2104 21 1068 8

6100 0.2125 20 1060 86200 0.2145 19 1052 76300 0.2164 18 1045 8

6400 0.2182 17 1037 76500 0.2199 17 1030 76600 0.2216 16 1023 7

6700 0.2232 16 1016 66800 0.2248 14 1010 7

6900 0.2262 14 1003 6

7000 0.2276 13 997 6

=

_



62 DIRECT FIRE

TABLE B


V 2300 f.s.

~_=~10gB' .1. u .1• T' .1• log C' .1.

--------- -- ----- --.--10

100 0.0036 35 2272 28 0.044 44 4.7878 35200 0.0071 36 2244 28 0.088 45 4.7913 35300 0.0107 36 2216 27 0.133 45 4.7948 36

400 0.0143 36 2189 27 017846 4.7984 36500 0.0179 36 2162 27 0.224 47 4.8020 36

600 0.0215 36 2135 27 0.271 47 4.8056 37

700 0.0251 36 2108 26 0.318 48 4.8093 37800 0.0287 37 2082 26 0.366 48 4.8130 37900 0.0324 37 2056 26 0.414 49 4.8167 '37

1000 0.0361 37 2030 26 0.463 50 4.8204 381100 0.0398 37 2004 25 0.513 50 4.8242 381200 0.0435 37 1979 25 0.563 51 4.8280 38

1300 0.0472 37 1954 25 0.614 51 4.8318 38~400 0.0509 37 1929 25 0.665 52 4.8356 391500 0.0546 38 1904 25 0.717 53 4.8395 39

1600 0.0584 37 1879 24 0.770 54 4.8434 391700 0.0621 38 1855 24 0.824 54 4.8473 391800 0.0659 38 1831 24 0.878 55 4.8512 40

1900 0.0697 38 1807 23 0.933 56 4.8552 402000 0.0735 38 1784 23 0.989 56 4.8592 402100 0.0773 38 1761 23 1.045 57 4.8632 41

2200 0.0811 38 1738 23 1.102 58 4.8673 402300 0.0849 38 1715 22 1.160 59 4.8713 412400 0.0887 38 1693 22 1.219 60 4.8754 42

2500 0.092538 1671 22 1.279 60 4.8796 412600 0.0963 39 1649 22 1.339 61 4.8837 422700 0.1002 38 1627 21 1.400 62 4.8879 42

2800 0.1040 38 1606 21 1.462 63 4.8921 422900 0.1078 39 1585 21 1.525 63 4.8963 423000 0.1117 38 1564 21 1.588 64 4.9005 42

3100 0.1155 37 1543 20 1.652 65 4.9047 433200 0.1192 38 152320 1.717 66 49090 433300 0.1230 38 1503 20 1.783 67 4.9133 43

3400 0.1268 38 1483 19 1.850 68 4.9176 443500 0.1306 38 1464 19 1.918 69 4.9220 443600 0.1344 38 1445 19 1.987 70 4.926.'1 44

=



D IRECT FIRE 63

TABLE B

(Taken from Ingalls' Table II, Artillery CircUlar M)

V 2300 f.s.

Xlog B' T' log C' ~z= - ~z U ~z ~zC

-103700 0.1382 38 1426 19 2.057 70 4.9308 443800 0.1420 38 1407 18 2.127 71 4.9352 443900 0.1458 38 1389 18 2.198 73 4.9396 45

4000 0.1496 38 1371 18 2.271 73 4.9441 454100 0.1534 38 1353 18 2.344 74 4.9486 454200 0.1572 38 1335 17 2.418 76 4.9531 45

4300 0.1610 37 1318 17 2.494 76 4.9576 45, 4400 0.1647 36 1301 16 2.570 77 4.9621 46

4500 0.1683 36 1285 16 2647 78 4.9667 46

4600 0.1719 35 1269 15 2.725 79 4.9713 464700 0.1754

35 1254 15 2.804 81 4.9759 464800 0.1789 34 1239 14 2.885 81 4.9805 46

4900 0.1823 34 1225 13 2.966 82 4.9851 465000 0.1857 34 1212 13 3.048 83 4.9897 465100 0.1891 33 1199 12 3.131 84 4.9943 47

5200 0.1924 32 1187 12 3.215 85 4.9990 465300 0.1956 31 1175 12 3.300 86 5.0036 475400 0.1987 30 1163 12 3.386 87 5.0083 46

5500 0.2017 29 1151 11 3.473 87 5.0129 475600 0.2046 28 1140 10 3.560 88 5.0176 465700 0.2074 26 1130 11 3.648 89 5.0222 47

5800 0.2100 26 1119 10 3.737 90 5.0269 465900 . 0.2126 25 1109 9 3.827 90 5.0315 476000 0.2151 24 1100 9 3.917 91 5.0362 47

6100 0.2175 23 10919 4.008 92 5.0409 476200 0.2198 22 1082 9 4.100 93 5.0456 46

6300 0.2220 22 1073 8 4.193 94 5.0502 46

6400 0.2242 QO 1065 8 4.287 94 5.0548 '466500 0.2262 20 1057 8 4.381 95 5.0594 456600 0.2282 19 1049 8 4.476 95 5.0639 46

6700 0.2301 18 1041 7 4.571 96 5.0685 456800 0.2319 17 1034 7 4.667 97 5.0730 446900 0.2336 16 1027 7 4.764 98 5.0774 45

7000 0.2352 15 1020 6 4.862 99 5.0819 45

=

'





DIRECT FIRE 65

TABLE B

(Taken froIn .Ingalls' Table II, Artillery Circular M)

V 2400 f.s.

Xz= - log B' ..1. u ..1. T' ..1• log C' ..1.

----10

3700 0.1372 38 1494 19 1.966 67 4.8920 443800 0.1410 39 1475 20 2.033 69 4.8964 443900 0.1449 38 1455 19 2.102 69 4.9008 44

4000 0.1487 38 1436 19 2.171 70 4.9052 444100 0.1525 38 1417 19 2.241 70 4.9096 454200 0.1563 38 1398 19 2.311 72 4.9141 45

4300 0.1601 37 1379 18 2.383 73 4.9186 454400 0.1638 37 1361 17 2A56 74 4.9231 454500 0.1675 37 1344 17 2.530 75 4.9276 45

4600 0.1712 37 1327 17 2.605 75 4.9321 464700 0.1749 37 1310 16 2.680 77 4.9367 464800 0.1786 36 1294 16 2.757 78 4.9413 46

4900 0.1822 36 1278 15 2.835 7f) 4.9459 475000 0.1858 36 1263 15 2.914 80 4.9506 475100 0.1894 36 1248 15 2.994 81 4.9553 46

5200 0.1930 35 1233 14 3.07.5 -81 4.9599 475300 0.1965 34 1219 13 3;156 83 4.9646 475400 0.1999 33 120613 3.239 83 4.9693 47

5500 0.2032 32 1193 13 3.322 84 4.9740 475600 0.2064 31 1180 12 3.406 85 4.9787 475700 0.2095 30 1168 12 3.491 86 4.9834 .47

5800 0.2125 29 1156 11 3.577 87 4.9881 485900 0.2154 28 1145 11 3.664 88 4.9929 476000 0.2182 27 1134 10 3.752 89 4.9976 47

6100 . 0.2209 26 1124 10 3.841 89 5.0023 486200 0.2235 26 1114 -10 3.930 90 5.0071 476300 0.2261 25 1104 9 4.020 91 5.0118 47

6400 0.2286 24 1095 9 4.111 92 ~5.0165 476500 0.2310 23 1086 9 4.203 92 5,0212 466600 0.2333 22 1077 8 4.295 93 5.0258 47

6700 0.2355 21 1069 8 4.388 94 5.0305 476800 0.2376 20 10618 4.482 95 5.0352 466900 0.2396 19 1053 8 4.577 95 5.0398 46

7000 0.2415 18 1045 7 4.672 96 5.0444 46

=

'

-

-

-



66 DIRECT FIRE

TABLE B

(Taken from Ingalls' Table II, Artillery Czrcular M)

V 2500 r.s.

33

222

X log B' t. z T' ~z log C' li z= - t. z uC

-10100 . 0.0035 35 2470 29 0.040 41 4.7150 35200 0.0070 35 2441 29 0.081 41 4.7185 35300 0.0105 35 2412 29 0.122 42 4.7220 35

400 0.0140 35 2383 29 0.164 42 4.7255 35500 0.0175 35 2354 28 0.206 43 4.7290 35600 0.0210 35 2326 28 0.249 43 4.7325 36

700 0.0245 36 2298 28 0.292 44 4.7361 36800 0.0281 35 2270 28 0.336 44 . 4.7397 36900 0.0316 36 2242 28 0.380 45 4.7433 . 37

1000 0.0352 36 2214 27 0.425 45 4.7470 371100 0.0388 36 21&7 27 0.470 46 4.7507 37

1200 0.0424 36 2160 27 0.516 47 4.7544 37

1300 0.0460 36 2133 27 0.563 47 4.7581 371400 0.0496 36 2106 26 0.610 48 4.7618 381500 0.0532 36 2080 26 0.658 48 4.7656 38

1600 0.0568 37 2054 26 0.706 49 4.7694 381700 0.0605 37 2028 26 0.755 50 4.7732 381800 0.0642 37 2002 26 0.805 50 4.7770 39

1900 0.0679 37 1976 25 0.855 51 4.7809 392000 0.0716 37 1951 25 0.906 52 4.7848 392100 0.0753 38 1926 25 0.958 52 4.7887 39

2200 0.0791 37 1901 25 1.010 53 4.7926 402300 0.0828 38 1876 24 1.063 54 4.7966 402400 0.0866 38 1852 24 1.117 54 4.8006 40

2500 0.0904 38 1828 241.171 55 4.8046

402600 0.0942 38 1804 23 1.226 . 56 4.8086 . 412700 0.0980 38 1781 23 1.282 57 4.8127 41

2800 0.1018 38 1758 23 1.339 57 4.8168 412900 0.1056 38 1735 23 1.396 58 4.8209 413000 0.1094 38 1712 22 1.454 59 4.8250 41

3100 0.1132 38 1690 22 1.513 59 4.8291 43200 0.1170 38 1668 22 1.572 60 4.8333 43300 0.1208 38 1646 21 1.632 61 4.8375 4

3400 0.1246 38 1625 21 1.693 62 4.8417 43500 0.1284 38 1604 21 1.755 63 4.8460 43600 0.1322 39 1583 21 1.818 64 4.8503 4

=





68 DIRECT FIRE

TABLEB


V 2600 f.s.

--'-

100 0.0034 35 2569 30 0.039200, 0.0069 34 2539 30 0.078

.300 0.0103 35 2509 30 0.118

400, 0.0138 35 2479 29 0.158

500 0.0173 35 2450 29 0.199600 0.0208 35 2421 29 0.240

700 0.0243 35 2392 29 0.281800 0.0278 35 2363 29 0.323900 0.0313 35 2334 . 28 0.366

1000 0.0348 35 2306 28 0.409.1100 . 0.0383 36 2278 28 0.4531200 0.0419

36 2250 27 0.4971300 0.0455 35 2223 28 0.5411400 0.0490 36 2195 .27 0.5861500 0.0526 36 2168 27 0.632

1600 0.0562 37 2141, 26 0.679'1700 0.0599 36 2115 27 0.7261800 0.0635 36 2088 26 0.773

1900 0.0671 37 2062 26 0.8212000 0.0708 37 2036 26 0.8702100 0.0745 37 2010 26 0.919

2200 0.0782 37 1984 25 0.9692300 0.0819 37 1959 25 1.0202400 0.0856 38 1934 25 1.071 .

2500 0.0894 37 1909 24 1.1232600, 0.0931 37 1885 25 1.1762700 0.0968 38 1860 24 1.230

2800 0.1006 37 1836 24 ,1.2842900 0.1043 38 1812 23 1.3393000 0.1081 38 1789 23 .1.394

3100 0.1119 38 1766 23 1.4503200 0.1157 38 1743 23 1.5073300 0.1195 38 1720 23 1.565

3400 0.1233 38 1697 22 1.6243500, 0.1271 38 1675 22 1.6833600 0.1309 39 1653 21 1.743

414242

383939

3737

38

404040

424243

38

3838

36

37

36

363636

35

3535

3435

34

393939 .

, 41

4141

!

log C'

-104.68094.68434.6878

4.6912

4.69474.6982

4.70174.70534.7089

4.71254.7161

4.7198

4.7234,4.72714.7308

4.73464.7~844.7422

9 4.74609 4.74980 4.7537

-1 4.75761 4.76152 4.7654

3 4.76934 4.77334 4.7773

5 4.78135 4.78546 4.7895

7 4.79368 4.79779 4.8019

9 4.806160 4.810361' 4.8145

5

555

555

555

445

424343

454647

474748

41

4141

4444

44

39

4040

5,5

5

T',x

Z=c C- log B'

=

'



))IRECT FIRE '69

TABLE B

(Taken from Ingalls' Table II,AriilleryCircu[ar M)

V 2600 f.~.

X. log B' ,T' " log C' .1,==- .1 z u .1 z .1 z

C--- --

-103700 0.1348 38 1632 22 1.804 62 4.8188 433800 0.1386 39 1610 21 1.866 63 4.8231 433900 0.1425 38 1589 21 1.929 63 4.8274 43

4000 0.1463 38 1568 21 1.992 64 4.8317 434100 0.1501 39 1547 20 2.056 65 4.8360 444200 0.1540 38 1527 20 2.121 66 4.8404 44

4300 0.1578 38 1507 19 2.187 67 4.8448 444400 0.1616 38 1488 20 2.254 67 4.8492 454500 0.1654 38 1468 19 2.321 69 4.8537 45

4600 0.1692 38 J449 19 2.390 70 4.8582 454700 0.1730 37 1430 19 2.460 70 4.8627 45

4800 0.1767 38 1411 18 2.530 72 4.8672 46

4900 0.1805 37 1393 18 2.602 72 4.8718 465000 0.1842 37 1375 18 2.674 73 4.8764 465100 0.1879 36 1357 18 2.747 74 4.8810 46

5200 0.1915 37 1339 17 2.821 75 4.8856 465300 0.1952 37 1322 17 2.896 76 4.8902 475400 0.1989 36 1305 17 2.972 77 4.8949 47

-5500 0.2025 36 1288 16 3.049 78 4.8996 475600 0.2061 36 1272 15 3.127 79 4.9043 475700 . 0.2097 36 1257 14 3.206 80 4.9090 47

5800 0.2133 35 1243 14 3.286 81 4.9137 485900 0.2168 35 1229 14 3.367. 82 4.9185 486000 0.2203 34 1215 13 3.449 83 . 4.9233 48

6100 0.2237 33 1202 13 3.532 84 4.9281 49

6200 0.2270 32 1189 12 3.616 84 4.9330 486300 0.2302 31 1177 12 3.700 86 4.9378 48

6400 0.2333 30 1165 12 3.78e -86 4.9426 486500 0.2363 30 1153 11 3.872 87 4.9474 496600 0.2393 28 1142 10 3.959 88 4.9523 48

. 6700 0.2421 27 1132 11 4.047 89 4.9571 486800 0.2448 26 1121 10 4.136 90 4.9619 496900 0.2474 26 1111 9 4.226 90' 4.9668 48

7000 0.2500 25 1102 9 4.316 91 4.9716 48

==



70 DIRECT FIRE

TABLE B

(Taken from Ingalls' Table II, AJ tillery Circular M)

V 2700 f.s.

T' Az log C' Az

-100.037 38 4.6480 340.075 38 4.6514 340.113 39 4.6548 34

0.152 39 4.658. 350.191 40 4.6617 350.231 40 4.6652 35

0.271 40 4.6687 35

0.311 41 4.6722 350.352 41 4.6757 36

0.393 42 4.6793 360.435 42 4.6829 36

0.477 43 4.6865 37

0.520 44 4.6902 360.564 44 4.6938 370.608 45 4.6975 37

0.653 45 4.7012 370.~98 46 4.7049 370.744 46 4.7086 37

0.790 47 4.7123 380.837 48 4.7161 . 37

0.885 48 4.7198 38

0.933 49 4.7236 380.982 . 49 4.7274 381.031 50 4.7312 39

1.081 51 4.735139

1.132 51 4.7390 391.183 52 4.7429 40

1.235 52 4.7469 401.287 53 4.7509 401.340 54 4.7549 41

1.394. 54 4.7590 411.448 55 4.7631 411.503 56 4.7672 42

1.559 57 4.7714 421.616 58 4.7756 421.674 58 4.7798 43

303030

302929

2929

28

282828

272727

-272626

262626

252525

252424

242424

232322

313131

u A

266926382607

2576254~2516

248624562427

23982369

2340

231222842256

22282201 .

6 2174

6 21476 21206 2094

7 20686 20427 2016

7 19907 19657 1940

7 19158 18908 1866

8 18429 18188 1794

8 17708 17478 1724

333

333

333

353636

333

3535

35

353535

3636

3

333

3435

34

343434

. 333

100 0.0034200 0.0068300 0.0102

400 0.0136500 '0.0170600 0.0205

700 0.0239800 0.0274900 0.0309

1000 0.03441100 0.0379

1200 0.0414

1300 0.04491400 0.04841500 0.0520

1600 0.05561700 0.05921800 0.0628

1900 0.06642000 0.07002100 0.0736

2200 0.07722300 0.08092400 0.0845

2500 0.08822600 0.09192700 0.0956

2800 0.09932900 0.10303000 0.1068

3100 0.11063200 0.11443300 0.1183

3400 0.12213500 0.12593600 0.1297

xz= - log B' AzC

=

'







DIRECT FIRE

TABLEB

(Taken frotnJngalls' Table II, Artillery Circularl"!)

V. 2800 f.s.

73

X log B'. T' ~z=-. Az u: AzC

3700 0.1319 38 1774 23 1.666 573800 0.1357 38 1751 23 1.723 573900 0.1395 38 1728 23 1.780 58

4000 0.1433 38 1705 22 1.838 59

4100 . 0.1471 . 38 1683 22 1.897 604200 0.1509 38 1661 21 1.957 60

4300 0.1547 38 1640 21 2.017 614400 0.1585 38 1619 21 2.078 634500 0.1623 38 1598 21 2.141 63

4600 0.1661 38 1577 21 2.204 644700 0.1699 38 1556 . 21 2.268 654800 01737 39 1535 20 2.333 66

4900 0.1776 . 38 1515 20 2.399 66

5000 0.1814 38 1495 20 2.465 675100 0.1852 37 1475 20 2.532 68

5200 0.1889 38 1455 19 2.600 695300 0.1927 37 1436 19 2.669 70'5400 0.1964 38 1417 18 2.739 '71

5500 0.2002 37 1399 1S" 2.810 725600 0.2039 38 1381 18 2.882 735700 0.2077 37 1363 18 2.955 74

5800. 0.2114 38 1345 17 3.029 755900: 0.2152 37 1328 16 3.104 766000 0.2189 38 1312 17 3.180 77

6100 0.2227 37 . 1295 16 3.257 786200 0.2264 36 1279 15 3.335 796300, 0.2300 36 1264 15 3.414 79

6400: 0.2336 35 124~ 15 3.493 806500 0.2371 34 1234 14 3.573 816"600' 0.2405 33 1220 14 3.654 83

6700~ 0.2438 33 1206 13 3.737 836800. 0.2471 32 1193 12 3.820 846900: 0.2503 31 1181 12 3.904 . 85

7000 0.2534 31 1169 11 3.989 86

log C'

-104.7507

.4.75494.7591

4.7633

4.76754.7718

4.77614.78044.7848

4.78924.79364.7980

4.80254.80704.8115

4.81604.82064.8252

482984.83444.8391

4.84384.84854.8533

4.85804.86284.8676

4.87244.87724.8821

4.88704.89194.8968

4.9017

424242

42

43.43

434444

44

4445

4545

45

46.4646

464747

474847

484848 .

4849

49

494949

50

=



74 DIRECT FIRE

TABLE B


V 2900 f.s.

X log B' Az Az T' Az log C' AzZ=c u

-10100 0.0032 33 2867 32 0.035 35 4.5857 33200 0.0065 33 2835 32 0.070 35 4.5890 33300 0.0098 33 2803 32 0.105 36 4.5923 33

400 0.0131 33 2771 32 0.141 36 4.5956 34

500 0.0164 34 2739 31 0.177 37 4.5990 34600 . 0.0198 34 2708 32 0.214 37 4.6024 34

700 0.0232 34 2676 31 0.251 38 4.6058 34800 0.0266 34 2645 30 0.289 38 4.6092 34900 0.0300 34 2615 31 0.327 38 4.6126 35

1000 0.0334 34 2584 30 0.365 39 4.6161 351100 0.0368 35 2554 30 0.404 39 4.6196 35

1200 0.0403 35 .2524 30 0.443 . 40 4.6231 36

1300 0.0438 35 2494 30 0.483 41 4.6267 351400 0.0473 35 2464 2\) 0.524 41 4.6302 361500 0.0508 35 2435 30 0.565 41 4.6338 36

1600 0.0543 35. 2405 29 0.606 42 4.6374 361700 0.0578 35 &2376 28 0.648 42 4.6410 371800 0.0613 . 35 2348 29 0.690 43 4.6447 36'

1900 0.0648 35 2319 . 28 0.733 43 4.6483 372000 0.0683 35 2291 28 0.776 44 4.6520 372100 0.0718 36 2263 28 0.820 44 4.6557 37

2200 0.0754 35 2235 27 0.864 45 4.6594 .382300 . 0.0789 36 2208 27 0.909 46 4.663'2 382400 0.0825 36 2181 27 0.955 46 4.6670 38

2500 0.0861 36 2154 27 1.001 47 4.6708 382600 0.0897 36 2127 27 1.048 47 4.6746 38

. 2700 0.0933 37 2100 26 1.095 48 4.6784 39

2800 0.0970 36 2074 26 1.143 49 4.1>823 392900 0.1006 37 2048 26 1.192 49 4.6862 393000 0.1043 37 2022 ,.26 1.241 50 4.6901 39

3100 0.1080 37 1996 25 1.291 50 4.6940 403200 0.1117 37 1971 25 1.341 51 4.6980 40

3300 0.1154 37 1946 25 1.392 52 4.7020 40

3400 0.1191 37 1921 25 1.444 52 4.7060 413500 0.1228 38 1896 25 1.496 53 4.7101 413600 0.1266 38 1871 24 1.549 54 4.7142 41

=

-



D IRECT FIRE 75

TABLE B

(Taken from Ingalls' Table II, Artillery Circular M)•V 2900 f.s.

Xlog B' T' Jog C' .1 zz= - .1 z u .1 z .1 z

C

-103700 0.1304 37 1847 24 1.603 55 4.7183 413800 0.1341 ~8 1823 24 1.658 55 4.7224 41

3900 0.1379 38 1799 23 1.713 56 4.7265 42

4000 0.1417 38 1776 23 1.769 57 4.7307 424100 0.1455 39 1753 23 1.826 57 4.7349 424200 0.1494 38 1730 22 1.883 58 4.7391 43

4300 0.1532 38 1708 22 1.941 59 4.7434 434400 0.1570 38 1686 22 2.000 60 4.7477 434500 0.1608 38 1664 22 2.060 61 4.7520 43

4600 0.1646 38 1642 22 2.121 61 4.7;)63 444700 0.1684 38 1620 21 2182 62 4.7607 444800 0.1722 38 1599 21 2.244 63 4.7651 44

4900 0.1760 38 1578 21 2.307 64 4.7695 445000 0.1798 37 1557 21 2.371 64 4.7739 455100 0.1835 37 1536 20 2.435 65 4.7784 45

5200 0.1872 38 1516 20 2.500 67 4.7829 455300 0.1910 37 1496 19 2.567 67 4.7874 455400 0.1947 38 1477 19 2.634 68 4.7919 46

5500 0.1985 38 1458 19 2.702 69 4.7965 465600 0;2023 38 1439 19 2.771 70 4.8011 465700 0.2061 38 1420 19 2.841' 71 4.8057 46

5800 0.2099 38 1401 18 2.912 72 4.8103 475900 0.2137 38 1383 18 2.984 73 48150 476000 0.2175 38 1365 18 3.057 74 4.8197 47

6100 0.2213 37 1347 18 3.131 74 4.8244 486200 0.2250 37 1329 17 3.205 76 4.8292 476300 0.2287 36 1312 16 3.281 77 4.8339 48

6400 0.2323 36 1296 16 3.358 77 4.8387 486500 0.2359 36 1280 15 3.435 79 4.8435 496600 0.2395 35 1265 15 3.514 79 4.8484 48

6700 0.2430 35 1250 15 3.5~3 80 4.8;)32 496800 0.2465 35 1235 14 3.673 82 4.8581 49

6900 0.2500 34 1221 13 3.755 82 4.8630 49

7000 0.2534 33 1208 13 3.837 83 4.8679 49

=

'

"



76 DIRECT FIRE

TABLE B

(Taken from lngalls~ Table II, Artillery Circular M)

V 3000 f.s.

log C' .1.

-104.5564 324.5596 334.5629 33

4.5662 334.5695 334.5728 34

4.5762 334.5795 334.5828 34

4.5862 33

4.5895 344.5929 34

4.5963 344.5997 354.6032 35

4.6067 35'4.6102 354,6137 36

4.6173 364.6209 364.6245 37

4.6282 ,374.6319 374.6356 38

4.6394 384.6432. 384.6470 38

4.6508 394.6547 394.6586 39

4.6625 40

4.6665 404.6705 40

4.6745 404.6785 404.6825 40

2 5051

.52

T' .1

0.0330.0670.101

0.1360.1710.207

0.2430.2790.316

0.353

0.3900.428

0.4670.5060.545

0.5850.6250.666

0.7070.7490.791

8 0.8348 0.8778 0.921

8 0.9667 1.0117 1.057

7 1.1036 1.1506 1.197 .

6 1.245

6 1.2936 1.342

5 1.392

25\ 1.442 I5 1.493

2 48

2 492 50

2 452 46

2 46

2 472 472 48

29 4229 4228 43

30 4029 4129 41

31 3930 3930 40

2 432 442 45

33 3433 3433 35

31 37

31 3831 39

31 3632 3731 37

32 3532 3632 36

X10gB'= - .1. uC

100 0.0031 32 2967200 0.0063 32 2934300 0.0095 32 2901

4000.0127 33 2868500 0.0160 33 2836

600 0.0193 33 2804

700 0.0226 34 2772800 0.0260 34 2741900 0.0294 34 2709

1000 0.0328 35 26781100 0.0363 36 26471200 0.0399 36 2616

1300 0.0435 36 25851400 0.0471 36 25541500 0.0507 36 2524

1600 0.0543 36 24941700 0.0579 36 24641800 0.0615 35 2435

1900 0.0650 36 24062000 0.0686 35 23772100 0.0721 36 2348

2200 0.0757 35 23202300 0.0792 36 22922400 0.0828 36 2264

2500 0.0864 35 22362600 0.0899 36 22082700 0.0935 36 2181

2800 0.0971 36 21542900 0.1007 36 21273000 0.1043 35 2101

3100 0.1078 36 20753200 0.1114

35 20493300 0.1149 37 2023

3400 37 19973500 37 19723600 38 1947

=

'

---- ---



DIRECT FIRE 77

TABLE B


V 3000 f.s.

Xlog B' T' log C' .1,= - .1 z u .1, .1,

C

-103700 0.1298 38 1922 24 1.545 52 4.6865 403800 0.1336 37 1898 25 1.597 53 4.6905 413900 0.1373 38 1873 24 1.650 54 4.6946 41

4000 0.1411 37 1849 24 1.704 55 4.6987 424100 0.1448 37 1825 24 1.759 55 4.7029 424200 0.1485 38 1801 24 1.814 56 4.7071 42

4300 0.1523 37 1777 23 1.870 57 4.7113 424400 0.1560 37 1754 23 1.927 57 4.7155 . 434500 0.1597 38 1731 23 1.984 58 4.7198 43

4600 0.1635 38 1708 22 2.042 59 4.7241 434700 0.1673 38 1686 22 2.101 60 4.7284 43

4800 0.1711 . 37 1664 22 2.161 60 4.7327 44

4900 0.1748 38 : 1642 21 2.221 61 4.7371 445000 0.1786 38 1621 21 2.282 62 4.7415 455100 0.1824 31' 1600 21 2.344 63 4.7460 45

5200 0.1861 38 1579 21 2.407 64 4.7505 455300 0.1899- 38 1558 20 2.471 64 4.7550 455400 0.1937 37 1538 20 2.535 66 4.7595 45

5500 0.1974 37 1518 20 2.601 67 4.7640 465600 0.20il 38 1498 20 2.668 67 4.7686 465700 0.2049 37 1478 19 2.735 68 4.7732 46

5800 0.2086 38 1459 19 2.803 69 4.7778 455900 0.2124 38 1440 19 2.872 70 4.7823 456000 0.2162 38 1421 19 2.942' 70 4.7868 45

6100 0.2200 39 1402 18 3.012 71 4.7913 45

6200 0.2239 39 1384 18 3.083 72 4.7958 . 456300 0.2278 39 1366 17 3.155 73 4.8003 46

6400 0.2317 38 1349 17 3.228 74 4.8049 4765(0 0.2355 37 1332 17 3.302 75 4.8096 476600 0.2392 37 1315 16 3.377 77 4.8143 48

6700 0.2429 36. 1299 16 .3.454 78 4.8191 496800 0.2465 35 1283 16 3.532 78 4.8240 506900 0.2500 34 1267 15 3.610 80 4.8290 50

7000 0.2534 34 1252 15 3.690 81 4.8340 51

'

==

-- --- --

'

'

'

'



78 D IR EC T FInE

TA B L E . C(From Ballistics, Part I, Hamilton)

Values of the coefficient of form, for projectiles the headsof which are of ogival radius n calibers.

n234567

c1.000.820.710.640.580.54

TA B L E D(From Ballistics, Part I, Hamilton)

Values of D.. in degrees, for use with the formula,

Deflection for W mile wind ~z ."Dw0

V 1600 1700 1800 1900

Z 0 0 0 0

0000 .0055 .0055 .0055 .00541000 .0059 .0058 .0058 .00572000 .0061 .0060 .0061 .00603000 .0061 .0062 .0063 .00634000 .0060 .0062 .0063 .00655000 .0059 .0061 .0064 .00666000 .0057 .0061 .0065 .00667000 .0055 .0060 .0064 .00658000 .0054 .0059 .0062 .00649000 .0053 .0057

.0060 .006310000 .0052 .0056 .0059 .006211000 .0051 .0055 .0058 .006112000 .0051 .0054 .0058 .006113000 .00nO .0054 .0057 .006114000 .0050 .0053 .0057 .006015000 .0049 .0053 .0056 .005916000 .0049 .0053 .0056 .005917000 .0049 .0052 .0055 .005818COO .0049 .0052 .0055 .005819000 .0048 .0052 .0055 .005720000 .0048 .0051 .0054 .005721000 .0048 .0051 .0054 .0057

2000 2100

0

.0054 .0053

.0057 .0056

.0060 .0059

.0063 .0062

.0065 .0065

.0067 .0067

.0068 .0069

.0068 .0070

.0067 .0070

.0066 .0069.00f;6 .0069

.0065 .0068

.0064 .0067

.0063 .0067

.0062 .0066

.0062 .0065

.0062 .0065

.0061 .0064

.0061 .0064

.0060 .0063

.0060 .0063

.0060 .0062

=





80 DIRECT FIRE

TABLE D-Continued

V 3100 3200 3300

Z0000 .0046 .0045 .00451000 .0048 .0047 .00472000 .0051 .0050 .00503000 .0054 •0053 .0053 .4000 .0058 .0057 .00575000 .0062 .0061 .00606000 .0065 .0064 .00637000 .0069 .0068 .00678000 .0073 .0072 .00719000 .0077 .0076 .0075

10000 ..0079 .0079 .007811000 .0081 .0081 .008112000 .0082 .0082 .008313000 .0082 .0083 .008414000 .0083 .0084 .0085

15000 .0083 .0084 .008516000 .0084 .0085 .008617000 .0084 .0085 .0086

. 18000 .0084 .0085 .008719000 .0084 .0086 .008720000 .0084 .0086 .008721000 .0084 .0086 .0087

3400 3500 3600

0

.0044 .0044 .0043.0046 .0046 .0045.0049 .0048 .0047.0052 .0051 .0050..0056 .0055 .0053.0059 .0058 .0056.0062 .0061 .0060.0066 .0065 .0064.0070 .0069 .0068.0075 .0073 .0073.0078 .0077 .. 0077.0081 .0080 .0080.0083 .0083 .0083.0084 .0085 .0085.0085 .0086 .0086

.0086 .0087 .0088

.0086 .0087 .0088

.0087 .0088 .0089

.0088 .0089 .0090

.0088 .0089 .0090

.0089 .0090 .0091

.0089 .0090 .0091

TABLE E(From Ballistics, Part I, Hmhilton)

Table of Values of percentage changes in muzzle velocityof powders tested at 70° F. and fired with a magazine tempera-ture to F.

t V/V V for V2000 2250 2500

0° -.0323 -65 -73 -8110 0 -.0302 -60 -67 -7520° -.0275 -55' -62 -69.30° . -.0241 -48 -54 -6040° -.0199 -40 -45 -5050° -.0147 -29 -33 -3660° -.0082 -16 -18 -2070° -.0000 0 0 080° .0101 20 23 2590° .0228 46 52 58

100° .0387 77 87 96

° ° ° ° °

~ ~ =

-



DIRECT FIRE 81

(Computed from the formula

A~ .00867 { 2.032t_ 2.032 X 70}

.00867 { 2. 032t _ 4.73 }.)

TABLE K (From Ballistics, Part I, Hamilton). R 2

K [3.3333 10] R2 .2154 (1000)

in which K is the curvature in feet and R the range in yards.The curvature in feet for ranges at 1000 yards interval are

given in the following table.Cur:vature inFee~ .

Range Curvature A Range Curvature A-yds. feet feet . yds. feet feet1000 0.22 0.64 12000 31.02 5.38

2000 0.86 1.08 13000 36.40 5.823000 1.94 1.51 14000 42.22 6.254000 3.45 1.94 15000 48.47 6.675000 5.39 2.36 16000 55.14 7.116000 7.75 2.80 17000 62.25 7.547000 10.55 3.24 18000 69.79 7.978000 13.79 3.66 19000 77.76 8.409000 17.45 4.09 20000 86.16 8.83

10000 21.54. 4.52 21000 94.99 9.2611000 26.06 4.96 22000 104.25

• TABLE P (From Ballistics, Part I, Hamilton)Table of Values of P v

v Pv A lJ P

v A.

100 .018 26 1600 1.177 136200 .044 33 1700 1.313 144300 .077 41 1800 1.457 128400 .118 48 1900 1.585 129500 .166 55 2000 1.714 128600 .221 62 2100 1.842 129700 .283 70 2200 1.971 129800 .353 77 2300 2.100 128900 .430 85 2400 2.228 129

1000 .515 92 2500 2.357 1281100 .607 .99 2600 2.485 1291200 .706 107 2700 2.614 128

1300 .813114 2800 2.742 1291400 .927 121 2900 2.871 129

1500 1.048 129 3000 3.000

=

=

= - =

-



82 DIRECT FIRE

FOT oblique impact, a being the angle of incidence

(measured from the normal to the plate) the following tableof percentages to be subtracted from perforations at normal. impact for each value of a, is given:

TA B L E Q.(From Ballistics, Part I, Hamilton)

a % a %

0 025 65 0 30 8

10 1 35 1115 2 40 1520 4 45 19

TA B L E R.

(From Ordnance Pamphlet No. 1872)

Normal Muzzle Velocities ....

Os:'Cl)

0."0S~~& 833 917 980 1056 1148 1220 2000 2100 2150 2200 2250 2400 2600- - - --

- _.- -- -- --- -- --

0 27 30 32 34 38 40 65 67 69 71 72 77 8426 29 31 33 37 39 63 66 67 69 70 75 810 25 28 29 31 35 37 60 62 64 66 67 71 77 fTOb'dedoot,d0 23 26 27 29 32 34 55 58 59 61 62 66 71 from normal0 20 23 24 25 29 30 49 51 53 54 55 59 64 muzzle velo-0 17 19 20 21 24 25 41 43 44 46 46 49 53 city.0 13 14 15 16 18 19 30 32 33 34 34 36 390 7 8 8 9 10 10 17

18 18 19 19 20 220 0 0 0 0 0 0 0 0 0 0 0 0 00 9 9 10 11 12 13 21 22

1

22 23 24 26 28 }To.be added to0 19 21 22 24 26 28 46 48 50 50 52 56 60 normal muz-00 32 35 37 41 44 47 77 81 82 84 87 92 100 zle velocity.

~

-



4

8

00 90 81 0.00958 \ 78 0.00917\ U.UU~ ~

90 91 0:01135 0.01084 83 0.01036 79 0.00992 76 0.00951 72

,81 93 0.0122 2 88 0.01167. 84 81 0.01068 77 0.01023 7490 0.01115

,74 95 0.0131092

0.01251 86 0.01196 82 0.01145 78 0.01097 75

69 97 0.01400 0.01337 87 0.01278 83 0.01223 79 0.01172 76

0.01492 93 0.01424.

166 98 95 89 0.01361 85 0.01302 81 0.01248 77

.64 100 0.0158596 0.01513 91 0.01446 86 0.01383 83 0.01325 79

'64 101 0.01680 0.01604 92 0.01532 88 0.01466 84 0.01404 80

:65 103 0.0177699

0.01696 94 0.01550 86 0.01484 82101 0.01620 90168 105 0.01875

102 0.01790 96 0.01710 92 0.01636 88 0.01566 84

)73 107 0.01976 0.01886 97 0.01802 93 0.01724 89 0.01650 85

l80 109 0.02078 104 0.01983 99 0.01813 90 0.01'735 86106 0.01895 95

~89 111 0.02182108 0.02082 101 0.01990 96 0.01903 92 0.01821 88

100 113 0.02288 0.02183 103 0.02086 98 0.01995 93 0.01909 90

:>13 115 o 02396 1090.02286 .104 95 0.01999 91

0'02505 112 0.02184 100 0.02088328 117 114 0.02390 107 0.02284 102 0.02183 97 0.02090 93

745 120 0:02617 0.02497 109 0.02386 104 0.02280 99 0.02183 95

122 o 02731 116 0.02606 0.02278 96865 118 111 0.02490 106 0.02379 101987 124 0:02847

121 0.02717 113 0.02596 107 0.02480 103 0.02374 98

111 127 0.02965 0.02830 115 0.02703 110 '0.02583 105 0.02472 101

129 o 03086 1230.02945 0.02573 102238 126 117 0.02813 112 0.02688 107

367 132 0'03209128 0.03062 120 0.02925 115 0.02795 109 0.02675 104

499 134 0:03335 0.03182 122 0.03040 116 0.02904 111 0.02779 106

136 0.03463 1300.03304 108633 132 124 0.03156 118 0.03015 113 0.02885

769 139 o 03593135 0.03428 126 0.03274 120 0.03128 115 0 . .2993 110

908 142 0:03725 0.03554 129 0.03394 123 0.03243 117 0.03103 112

141 o 03860 138 0.03683050 0'03998 140 131 0.03517, 125 0.03360 120 0.03215 1140194 147

0:04138 143 0.03814 134 0.03642 128 0.03480 122 0.03329 116,341 150 0.03948 136 0.03770 130 0.03602 124 0.03445 11

153 0.04281 1460.04084l491

156 o 04427 1480.04223

139 0.03900 133 0.03726 127 0.03563 1211644

158 0:04575 1510.04365

142 0.04033 135 0.03853' 129 0.03684 121800 144 0.04168 138 0.03982 132 0.03808 12

L958 161 o 04726 1540.04509

0:04880 157 147 0.04306 140 0.04114 134 0.03934 12)119 164 160 0.04656 150

167 0.05037 0.04806 0.04446 143 0.04248 137 0.04062 13

)283 152 0.04589 146 0.04385 140 0:04193 13:>450 170 0.05197 162

0.04958173 0.05359 165 0.05113

155 0.04735 148 0.04525 142 0.043265620

176 0.05524 168 0:05271 158 0.04883 151 0.04667 144 0.04462~793 1()0 0.05034 153 0.04811 147 0.04600

5969 179 0.05692 171 0.05431164

5148 182 0.05863 174 0.05595167 0.05187 157 0.04958 149 0.04740

5330 185 0.06037 177 0.05762 169 0.05344 160 0.05107 152 0.048840.05504 162 0.05259 155 0.05030

188 0.06214 180 0.05931 1730.05666515

0.06394 183 0.06104 176 165 0.05414 158 0.051796703 191

0.06577186 0.06280 178 0.05831

169 0.05572161 0.05331

6894 194 0.06000 171 0.05733 164 0.05485

0.06763 '189 0.06458 1810.061717088 196 0.06639

17284 199 0.06952 192 0.06823184

0.06345 174 0.05897 167 0.05643

17483 202 0.07144 195 1870.06522 177 0.06064 170 0.05803

180 173 0.059660.07010 191

0.06234

17685 205 0.07339 197 0.07201 193 0.06702183 16

)7890 208 0.07536 201 0.07394 196 0.06885185

0.06407 176 0.06132

)8098 211 0.07737 204 0.07070 0.06583 179 0.06300 17

0.07590 199189 0.06762 182 0.06472 17

)8309 214 0.07941 206 0.07789 2020.07259

191)8523 217 0.08147 210 0.07450 0.06944 185 0.06646 17

)8740 220 0.08357 212 0.07991 205 0.07645 195 0.07129 188 0.06824 18197 0.07317 191 0.07004 180.08196 208 0.07842)8960 223 0.08569 215 211 201

)9183 226 0.08784 2190.08404

2140.08043

203 0.07508 193 0.0718709409 230 0.09003 221

0.08615 0.08246 206 0.07701 195 0.07373

0.08829 217 0.084520.07896 198 0.07562 H

09639 234 0.09224 221-- 209 0.08094:::=="' 201 0.07754

~ . ,

131314

14 1414

151515

161616

11

~









84 HIGH ANGLE FIRE

w is the weight of the projectile in pounds.d is the caliber of the projectile in inches.i is the index of the projectile and refers to its form.

For the ordinary or Johnson cap, its value is unity fOf thepurposes of this article, where the head has a two caliberradius of ogive; for the long pointed projectile, having anogival head struck with a radius of 7 calibers, and no cap, thevalue of i is for mortar fire 0.75.

The first value of i (that for the cap) was assumed at theoutset and the second was deduced from actual firings onthe basis of laws established with the capped projectile.

2. With the ordinarily assumed law of resistance for thevelocities usual in mortar fire, the retardation is representedby,

r=F~) .00004676~

V2[5.6699 -10] C

This refers to a two-caliber radius of ogive, without cap.The actual firings with the capped projectile gave,

F(Y) V2r =----c- =K C

in which

{I 100( V -800) }

K =.00005 1- 4 . 1002+ (V -800)2

{1100 (V 800) }

[5.6990 -10] 1 -4 . 1002+ (V _ 800)2

The values of log K, are given in Table II, page 98, forthe zone velocities. Its computation for any velocity presents

no difficulty.3. At the point of fall the relation between the range X

(feet) and the fundamental quantities was found to be,

V2 sin 2 cp 1 + 4 F ( V) , .gX 3. g c . 1\ SIn cp

in whichg is the accelleration due to gravity (= 32.16 f.s. per sec.)

and

, . 2.41\ sIn~ (1 +sin 2 cp)i

Values of log A sin cp log E are given in Table III.

=

=

- =

=

=

=



HIGH ANGLE FIRE

4. Representing by M the ratio,

gXV2 sin 2SO

and by N the ratio,

85

gT2 V sin SO

-in which T is the time of flight, in seconds, to the point offall, for which the range is X,-we have ~rom the firings,

1M=l +B

In which,.

B i .F:%) . Asin ~, for all projectiles

and N ==M(1 + i~3~t),or 1046-lb.

N 16.884 ( Si~t ) t for 824- and 700-lb.

5. The expressions for the remaining range-table ele-ments as deduced from the firings are given below:

R (d) R MV2 sin 2SO MV2 sin 2SOange yar s 3g 96.48

'. 2 N V sin SO N V sin SOTIme of FlIght (secon~s) T g.. 16.08

, d3

Drift (degrees) = 2 ( 1 - K ) - . SOOec SOwn

p D (Tables IV and V.)tan SO

tanw=~Velocity of Fall VW =M~ V cos SOec w

Thickness of Deck} =e=[5.4175-10j IWVW3{1_ (90-W)2}

Steel perforated ~. d 100

In these formulas M, N, w, d are as already defined.K is a drift constant having a value (for projectiles with

Johnson caps) t, and a value 0.837 for projectiles with 7-caliberradius of ogive.

n is a drift constant, defining the twist of rifling, andas the mortar, model 1890 (and subsequent models), has atwist of one turn in 20 calibers, n 20 for these. For the C.Lmortar, model 1886, and for the steel mortar, 1886-90, n = 25.

=

=

= =

= = =

=

=

=



86 . HIGH ANGLE FIRE

e is the thickness of deck steel perforated, and is in inches.Vw and ware the velocity of fall and the angle of fall, respec-tively. 'In the value of e the factor,

1_(9~~ow)2 .takes cognizance of oblique impact on the horizontal deck, due.to the angle of fall. Thus its value for a 60° angle of fall is,

1- (9~~60) 2 1-.09 =0.91

The drift is given in degrees: .The values of D.= cp sec cp

are tabulated and shown in Table IV, and those of p. d3

2 (1 K) are shown in Table V.wn

In Table II we shall give,

log F~~) =log K

log F(V) =log KV2. 4 F(V)

log 3. -g- =log H

and in Table I

log A sin cp log Eso that

B=HEC1

M=l+B

II . CORRECTIONS FOR ABNORMAL CONDITIONS

The range-tables as calculated above apply when normalconditions exist; that. is, .when the muzzle velocities and airdensity are those presumed in the range-table, and when thereis no motion of the atmosphere.

As these conditions are ideal, it is necessary to providemeans of adapting the elevations to actual conditions.

Formulas for this purpose are readily found as follows:

1 _ V2sin 2SO 1 + 4 F ( V) 1 •

M gX . 3' ..g c .1 \ sIn SO

4 V2K .1 +3" C" g " ASIn SO

=

=

- -

=

-

=



HIGH ANGLE FIRE

Simple differentiation gives,.~ (2dV + d sin 2cp _ dX)M V sin 2cp X

(-l_ 1) {2 d V _ dC + dK + dA s.in cp}M V C K A SIn cp.

Now. 2.4

. A SIn cp (l + sin 2 cpand hence

87

So that

d A sin cp 5 d sin 2 cp

A sin cp 3 . 1 + sin 2 cp

dX -2MdV _(l_M)dKX V K

.+ (l_M)dCC .

+. { 1 +~(1 _ M) sin.2cp } d .sin 2cp .3 1 + SIn 2 cp sm 2 cp

This formula affords the .means of making slight changes,but it is better to ascertain the effects by recalculating, using amodified V or C to accord with the facts.

Specifically, the abnormalities which, in practice, causecorrections to be necessary are:

1. Change in ballistic. condition of powder due tostorage.

2. Change in atmospheric density.3. Motion of the atmosphere.

1. POWDER CHANGES

These are rather complex; but the matter is. finally

summarized by saying that a change in powder conditionproducing a 1 per cent range effect, either with the 1046-lb.pr.ojectile at 1050 f.s., or with the 824-lb. projectile at 1300f.s., will produce changes in other zones as shown in Table Aon page 88.

I t is to be noted that the percentage range correctionhaving been determined for one zone, that for each of theothers may be found by proportion. This applies to the first7 zones and the zone with the 824-lb. projectile and muzzle

velocity 1300 f.s., when the same powder is employed through-out. As a different (coarser) powder. is employed with the700-lb. projectile, the zones using this projectile are not sorelated to the others.

=

= ~

= -

-

:/



88HIGIi ANGLE FIRE

Table A

Zone

123

4.5

678

v

550600660

.725810915

I. 1050

1300

!lR/R%

1.931.84 .1.801.801.701.401.001.00

2. ATMOSPHERE

dR de (al )

Ii ( l - M ) c = ( l - M ) T -1 .3. MOTION OF THE ATMOSPHERE

. Due to the great altitudes reached by mortar projectilesin their flight, the mean direction and rate of motion of theatmosphere as a whole can only be inferred from the fall of theshots. At the same time, it has been practicable to demon-strate by analytical methods that the deflection produced bya 10-mile-per-hour lateral movement of the atmosphere will,in the case of each of the three mortar projectiles, produce alateral motion on the part of the projectile, equivalent in effectto 8.7 per cent of the drift for that projectile. This is import-ant in that it permits the cross wind-component to be readilyinferred, and corrected by making a percentage correction in thedrift.

In regard to the effect, on the range, of motion of theatmosphere in the plane of fire, exhaustive discussion of thesubject shows .

a. That the range effect of wind is sensibly constant inany zone, between 45° and 650.

b. That in the case of the 12-inch mortar this effect inyards is given by,

JR=5W x{

106~rin which W x is the velocity in miles per hour of the at-mosphere in the direction of fire. Hence, for a 10-milecomponent,

=





90HIGH ANGLE FIRE

V 550 600 660 725 810 915 1050

-- -- -- -- -- ----I sin 2~og--

3g

log R65

R 65

7.89987.89987.89987.89987.89987.89987.89981--

3.34423.41273.4861 3.55753.64873.74533.83932209 2586 3063 3610 4453, 5563 6907

Ex. 2. The thermometer reads 79°, and the barometer29.40 inches; what effect will this have on the ranges?

Table I gives

Now,

and, in this case,

~=1.050(J

dJ: (l ~Jl;I)

915 1050.8634 .8215

.1366 .1785

.8370 .7890

.1630 .2110

dCC ~l -1 =.050

The values of M are the reciprocals of 1 +B, and a.c. log(1 + B) is log M. Consequently we have,

v 550 1 600 6601 725 810

M 45 .9339 .9213 .9049 .8874 .8788(l-M)45 .0661 .0787 .0951 .1126 .1212

M 65 .9198 .9049/ .8855 .8650 .8549(1 -M)65 .0802, .0951 .1145 .1350 .1151

Hence the percentage range changes are,

At 45 /.003311.003941.004761.005631.006061.006831.00893At 65

0

.00401.00476.00573.00675.00726.09815.01055

And the actual range changes ate,

. Ex. 3. Calculate the values of log H, and of log C for thefollowing zones for the 12 inch mortar, model 1890.

(a) V 1300

projectile 824-lb., Johnson cap(b) V 1250}(c) V 1500

projectile 700-lb., long pointed, 7 cal. rad. ogive

---- -

--------

= - ~

=

°

=

= =



HIGH ANGLE FIRE 91

(b) (c)

450 700202500 490000

10000 1000045000 70000

212500 50000018 7

. 85 509 7

170 200161 193170 200

9.9764 9.98456.1938 6.35225.6990 5.69908.6176 8.61760.4868 0.65332.8451 2.84517.8416 7.84160.1249 0.12490.8116 0.8116

.(a)

V-800 500(V -800)2 250000

100 2 10000100 (V -800) 50000

100 2 + (V -800)2 260000

ratio5

26

ratio5

104

1 ratio99

104log 9.9786log V2 6.2279

.c. log 20000 5.6990logUg 8.6176log H 0.5231log w 2.9159a.c. log d2 7.8416a.c. log i 0.0000log C 0.7575

a

Ex. 4. Given,V 1500cp 56°R 13875 yards700 lb. long pointed projectile

Calculate,a. Time of flight.b. D rift.c. Angle of fall.

d. Maximum ordinate.e. Velocity of fall.f. Thickness of deck armor perforated.

Here,

a. N 16.884 (Si~t) t

T=2NVsin~g

1.05 VO.6 (sin~)1.2

log 1.05 .02120.6 log V 1.90571.2 log sin cp 9.9023

-

-

-

~ -

- ~ -

= = =

=

=

= =



92

b.

log TT

HIGH ANGLE FIRE

1.8292

67.48 sec.

(1)

2 ,(1 K) = 2 X .163 = .326log .326 9.5132-10log d3 3.2375

a.c. log w 7.1549 -10a.c. log n 8.6990 -10

log rp see rp

log Drift 0.6052Drift 4°.029

(2)

log p 8.6046 -10 Table Ylog D 2.0006 Table IV

log Drift 0.6052Drift 4°.029

C.

Hence

(1)

tan w 1 y 2 sin 2rptan rp M 3gR

(2)

2 y2 sin 2 rptanw=---_

96.48 R2 log V2 log sin rp

a.c. log Ra.c. log 48.24

log tan ww

y2 sin 2 rp

48.24 R6.35229.83725.85788.3166

0.363866°36'

tan rptanw= -u

log 3glog R

a.C. 2 log Va.C. log sin 2rp

logMa.C. log M

log tan rp f.

log tan ww

1.98444.14223.64780.0328

9.80720.19280.17100.363866°36'

d. Yo tg T2

4.02 T2

from (a) 2 log T 3.6584

log 4.02 .6042

e.

log Yo= 4.2626Yo 18306 feet

v'" M3/2 Y cos rp see w3/2 log M = 9.7108-10

log Y 3.1761logcos rp 9.7476

a.C. log cos w 0.4010

log v'" 3.03551'", 1085 f.s.

-= = = = = ~ = =

= =

= =

= =

= = = = = =

= = = = = = = = =

= = =

=

=

=

= = =

= =



HIGH ANGLE FIRE 93

, WVw3 ff. e =[5.4175-10 l"-d-l 1

{jJ 66.690 -lJJ 23.4

(90 -lJJ).2

100 .055

1 (90 -lJJ ) 2100 .945

log W =2.84513 log Vw 9.1065

a.c. log d 8.9208 -10

(9~~OlJJ) }

con~t log 5.4175-10log .945 9.9754

2)10.8724log e 0.8291

e 6.747 inches

5.4362.

Ex. 5. Firing of trial shots in zone 5 (1046-lb., 810 f.s.)with barometer 29.4 and thermometer 79°, showed a range6100 yards at 45°. Atmosphere still and clouds motionless.

(a) What part of this was due to powder?(b) What percentage correction should be made for pow-

der in each zone?

(a) Ex. 2 shows that 36 yards of this was due to air den-

sity; so that the range, all conditions normal except powder,was 6100 36 6064.Ex. 1 shows that the range-table range is 5977 yards.

Hence the shot ranged too far by6064 5977 87 yards 1.45 %

Table A (page 88) shows 1.70 per cent as correspondingfor this zone to one per cent in the 7th and 8th zones. Hencethe percentage figures should be multiplied by

1.45 291.70 34

giving:

Zone I Effect Correction

---I 1.65 % -1.65 %2 1.56 % -1.56 %3 1.53 % -1.53 %4 1.53 % -1.53 %5 1.45 % -1.45 %6 1.19 % -1.19 %7 0.85 % -0.85 %8 0.85 % -0.85 %

-

= =

=

_ =

= =

= =

= =

- =

- = =

=



94HIGH ANGLE FIRE

Ex. 6. Shots were fired in the 6th zone (1046-lb. pro-

jectile, V 915 f.s.) at an elevation 590.Four shots were fired at azimuth 230° and four at azimuth

290°.

The mean ranges and deflections observed were:

At az. 230° At az. 2900Range 6628 yards 6670 yardsDeflectio:p. 4°.70 30.58

The barometer stood "at 28.50 and the thermometer at 700.The range table shows for this zone and elevation,

Range-6532 yardsDrift- 3°.79

(a) Find mean direction and rate of motion of the atmos-phere.

(b) Find the percentage range correction due to powder

in this and other zones.Let A be the azimuth of the direction towards which the

atmosphere is moving. Then A -180° will be the azimuthfrom which it comes, or its "azimuth."

Then if Wbe the rate of motion of the atmosphere in milesper hour, and JR v be the effect in yards on the range of thechange in condition of the powder, we shall have,

Az. 230° Az. 2900Deflecting componen t of W, W sin (A 230°) W sin (A _ 2900)Range component of W, W cos (A 230°) W cos (A _ 2900)or, representing A -290° by ex,

Deflecting componentw sin (a+600) W sin ex

Range component W cos (a+600)W cos aTo reduce these components to effects in yards, it is

necessary to note that:

The deflection due to a 10-mile wind is 8.7% of the drift.The range effect due to a 10-mile wind is given by

J R = 50 ( 1060 ) 2

= 50 (1~~~)2=. 42 yards in this case

The deflection for 10-miles per hour is,.087 X3.79

0°.33

=

- -





96HIGH ANGLE FIRE

-'-42 yards

These effects, in view of the fact that 1 mi. per hr. pro-

duces a range effect 4.2 yards, may be used as follows:JR v + 4.2 W cos (a + 60°) = 35JRv + 4.2 Wcos a 77

By subtraction,4.2 W {cos (a + 60°) -cos a}

or

W{cos (a+600) -('os a}

Now, a -10046'.a + 60° 49° 14'

cos (a + 60°) 0.6529cos a = 0.9824cos (a+600) cos a -0.3295

Hence,

-10 .

-10W= -0.3295

30.4 mi. per hour.JRv = 77 4.2 Wcos a

77 4.2 X 30.4 X 0.9824= 77 125

53 yardsHence the percentage change due to powder is,

-536532 = -0.81 %Hence, (a) Azimuth =

A-180° 290°+ a-180° 990W 30 mi. hr.

(b) Percentage change in range in this zone -0.81 %.Percentage change in other zones, from Table A:

Correctionffectone

I 1.93 X;'" 1~~ = -1.12~ +1.12%

II 1.84 X -1.07; +1.07%III 1.80 X = - .04; +1.04%IV 1.80 X = -1.04; +1.04%. V 1.70'X = ;"'0.98; +0.98%VI 1.40 X = -0.81; +0.81%

VII 1.00 X = -0.58; +0.58%

VIII 1.00 X -0.58; +0.58%

(REMARK: Had no allowance been made for wind it wouldhave been presumed that the average effect due to powder was,

=

= =

- =

= -

= --

= -

= = =

=

=



HIG~ ANGLE FIRE 97

35+77-2-.- +56 yards

and a percentage correction in the various zones in exactly the

opposite direction, and almost equal in amount would havebeen made, and in firing on a different day a large error wouldhave been made in powder correction with no assurance of .acompensating wind. Even on the same day, large errors wouldhave been made in other zones; and some error even in thiszone.)

TABLE I

Values of aI/a for temperature and pressure of atmosphere 78% saturated

with moisture. (From Artillery Note No. 25.).

her. I Barometer Ther. BarometerF. 28" 29" 30" 31" F. 28" 29" 30" 31"

0 0

20 0.890 0.861 0.831 0.806 9 0.950 0.887 0.887 0.85819 0.8~2 0.863 0.833 0.808 10 0.952 . 0.919 0.889 0.86018 0.894 0.864 0.835 0.809 11 0.954 0.921 0.890 0.86217 0.896 0.866 0.837 0.811 12 0.956 0.923 0.892 0.864

16 0.898 0.868 0.839 0.813 13 0.958 0.925 0.894 0.86615 0.901 0.870 0.841 0.815 14 0.960 0.927 0.897 0.86714 0.903 0.872 0.843 0.816 15 0.962 0.929 0.899 0.86913 0.905 0.874 0.845 0.818 16 0.964 0.931 0.901 0.87112 0.907 0.876 0.847 0.820 17 0.966 0.933 0.903 0.87311 0.910 0.878 0.848 0.822 18 0.968 0.935 0.905 0.87510 0.912 0.880 0.850 0.824 19 0.971 0.937 0.907 0.8779 0.914 0.881 0.852 0.826 20 0.973 0.939 0.909 0.8798 0.916 0.883 0.854 0.827 21 0.975 0.941 0.911 0.8817 0.918 0.885 0.856 0.829 22 0.977 0.943 0.912 0.8836 0.920 0.887 0.858 0.831 23 0.979 0.945 0.914 0.8855 0.922 0.889 0.860 0.833 24 0.981 0.947 0.916 0.8874 0.924 0.891 0.862 0.835 25 0.983 0.949 0.918 0.8883 0.926 0.893 0.864 0.836 26 0.985 0.951 0.920 0.8902 0.928 0.895 0.866 0.838 27 0.987 0.953 0~922 0.8921 0.930 0.897 0.868 0.840 28 0.990 0.955 0.924 0.8940 0.932 0.899 0.870 0.842 29 0.992 0.958 0.926 0.8961 0.934 0.901 0.871 0.844 30 0.994 0.960 0.928 0.8982 0.936 0.903 0.873 0.845 31 0.996 0.962 0.930 0.8993 0.938 0.905 0.876 0.847 32 0.998 0.964 0.932 0.9024 0.940 0.907 0.878 0.849 33 1.000 0.966 0.934 0.9035 0.942 0.909 0.880 0.851 34 1.003 0.968 0.936 0.9066 0.944 0.911 0.881 0.853 35 11.0050.970 0.938 0.907.7 0.946 0.913 0.883 0.855 36 1.007 0.972 0.940 0.9098 0.948 0.915 0.885 0.856 37 1.009 0.974 0.943 0.911

T

=



98HIGH ANGLE ,FIRE

Ther.Barometer Ther. BarometerF. 28" 29" 30" 31" F. 28" 29" 30" 31"

0

0

38 . 1.011 0.976 0.945 0.913 70 1.082 1.044 1.009 0.9779 1.013 0.978 0.947 0.915 71 1.085 1.046 1.011 0.9790 1.015 0.980 0.949 0.917 72 1.087 1.048 1.013 0.9811 1.017 0.982 0.951 0.919 73 1.089 1.050 1.015 0.9832 1.019 0.984 0.953 0.921 74 1.092 1.053 1.017 0.9853 1.021 0.9870.955 0.923 75 1.094 1.055 1.019 0.9874 1.023 0.989 0.957 0.925 76 1.096 1.057 1.022 0.9895 1.026 0.991 0.959 0.927 77 1.099 1.059 1.025 0.9926 1.028 0.993 0.961 0.929 78 1.101 1.062 1.027 0.9947 1.030 0.995 0.963 0.931 79 1.104 1.064 1.029 0.9968 1.033 0.997 0.964 0.933 80 1.106 1.066 1.031 0.9989 1.035 0.999 0.966 0.935 81 1.109 1.068 1.033 1.0000 1.037 1.002 0.968 0.937 82 1.111 1.071 1.035 1.0021 1.040 1.004 0.970 0.939 83 1.114 1.074 1.038 1.0052 1.042 1.006 0.972

0.941 84 1.116 1.076 1.041 1.0073 1.044 1.008 0.974 0.943 85 1.119 1.079 1.043 1.0094 1.046 1.010 0.976 0.945 86 1.121 1.081 1.045 1.0115 1.048 1.012 0.978 0.947 87 1.124 1.083 1.047 1.0136 1.050 1.014 0.980 0.949 88 1.126 1.086 1.049 1.0167 1.0.53 1.016 0.982 0.951 89 1.129 1.089 1.053 1.0188 1.055 1.018 0.984 0.952 90 1.131 1.092 1.055 1.0209 1.057 1.020 0.986 0.954 91 1.134 1.094 1.057 1.0220 '1.059 1.022 0.988 0.956 92 1.136 1.096 1.059 1.0251 1.062 1.025 0.990 0.95893 1.139 1.099 1.062 1.0272 1.064 1.027 0.992 0.960 94 1.142 1.102 1.064 1.0293 1.066 1.029 0.994 0.962 95 1.144 1.105 1.066 1.0314 1.068 1.031 0.996 0.964 96 1.147 1.107 1.068 1.0335 1.071 1.033 0.998 0.966 97 1.149 ~.11O 1.071 1.0356 1.073 1.035 . 1.001 0.968 98 1,152 1.112 1.074 1.0377 1.075 1.037 1.003 0.970 99 1.155 1.115 1.076 1.0408 1.078 1.040 1.005 0.973 100 1.157 1.117 1.079 1.0429 1.080 1.042 1.007 0.975 -- --- --- --- ---

TABLE IIV log K log F (V) log H

550 5.7349-10 1.2156 9.8332 -1000 5.7404-'10 1.2967 9.9143 -1060 5.7475-10 1.3866 0.004225 5.7482-10 1.4689 0.086510 5.6881-10 1.5051 0.122715 5.6416 -101.5644 0.1820050 5.6598-10 1.7022 0.3198250 5.6754-10 1.8692 0.4868300 5.6776-10 1.9055 0.5231500 5.6835-10 2.0357 0.6533



99

TABLE IVValues of log (I{J sec I{J)

log D

HIGH ANGLE FIRE

TABLE IIIValues of log E log (A s in rp).

logE Ll I{J

logELl

9.8785 -10 2 58 9.9161 -10 619.8787 7 59 9.9222 659.8794 11 60 9.9287 709.8805 15 61 9.9357 759.8820 20 62 9.9432 809.8840 25 63 9.9512 849.8865 29 64 9.9596 899.8894 33 65 9.9685 949.8927 37 66 9.9779 99

9.8964 42 67 9.9878 1049.9006 47 68 9.9982 1109.9053 52 69 0.0092 1169.9105 56 70 0.020~

I{J log rp sec I{J I q; log I{J sec I{Ji

5 1.8037 I 58 2.03926 1.8210

I59 2.0590

7 1.8383 60 2.07928 1.8558

I61 2.1000

9 1.8733 62 2.12090 I

1.8909 63 2.14231 1.9087 I 64 2.16442 1.9267I 65 2.18703 1.9448 66 2.2102

4 1.9632 67 2.2342I5 1.9818 I 682.25896 2.0006 69 2.2846

7 2.0198 I 70 2.3110

---

5

5

I{J

---5

4647

4849505152

5354

5556

----444455555

I

TABLE V. Values of log p

Projectiles

n 1046-1b., capped !_82_4 1_b_.,_c_a_ pp_e_d_I_7_0_0-_lb_._,lO_ll_g_P_Ol_'l_te_d

1890 20 8.5191-10 8.6257-10 8.6046-10

8.4222-10 8.5288-10 8.5077-10d3

p=2(l-K)-wn

Drift pD

=

= ~

'

- --

I

i

I I

~

_____

~ __

~

=



Greel{ Letters

A a AlphaB fl Betar r GammaJ DeltaE e EpsilonZ C ZetaH T) Etaf) {} ThetaI t Iota

K " KappaA A LamdaM p. MuN 1) Nu

Xi:f,

0 0 OmicronII 7T: Ptp p RhoX a Sigma

T r Taur u UpsilonrP cp PhiX I Chi1Jf cjJ Psi!J w Omega

~

""'" ~

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Notes on Ballistics