Simple MendelianPopulation Mating at Random. 37rspb.royalsocietypublishing.org/content/royprsb/83/561/37.full.pdf · See W. Weinsberg, i Zeitschrift fur Vererbungslehre ; 1909, vol

prove of considerable value in bacteriological investigations. Further experiments are now being made with other organisms which up to the present time have not been proved to be absolutely distinct.

Numerous experiments were made on the resistance of these endotoxins to heat, and every result shows that they are thermostable even when exposed to a temperature of 100° C. for 30 minutes, and are still able to exert their specific action, as is shown by the following experiment:—

Simple MendelianPopulation Mating at Random. 37

No of bacilli in 50 cells.

No. ofnon-phagocytic

cells.

A. Serum saline -f leucocytes + B. coli ........................................ 233 0B. Serum Danysz extract + leucocytes + B. coli . ...................... 59 15C. Serum coli extract 4- leucocytes + B . coli ............................... 4 46D. Serum Danysz extract* + leucocytes + B. co li ....................... 59 16E. Serum coli extract* + leucocytes + B. coli ........................... 6 44

* This extract was exposed to a temperature of 100° C. for 30 minutes.

On the Determination of the Chief Correlations between Collaterals in the Case of a Simple Mendelian Population Mating at Random.

By E. C. S n o w , B.A., Biometric Laboratory, University College, London.

(Communicated by Prof. Karl Pearson, F.R.S. Received June 3,—ReadJune 30, 1910.)

(1) In a paper communicated to the Royal Society in April, 1909, Prof. Pearson obtained the gametic correlations between the offspring and the ancestry in each grade in a simple Mendelian population mating at random. By a “ simple Mendelian population ” for a given character, he understood one which started with any definite ratio of dominant individuals (AA) to recedents ( aa). These mating at random give rise to a population which may be written in the form

p2 (AA) -(- 2 pq(A -+■ q2 (aa),and of which, without selection, the proportions of dominants, recedents, and hybrids are known to remain constant, with continued random mating, during successive generations. Prof. Pearson found that, both in the case of gametic and somatic characters, the ancestral correlations diminished in

on September 16, 2018http://rspb.royalsocietypublishing.org/Downloaded from

http://rspb.royalsocietypublishing.org/

38 Mr. E. G. Snow. Correlations between Collaterals [June 3,

a geometrical progression, and thus obeyed the fundamental principle of the Law of Ancestral Heredity, as deduced from observations on man and other living forms.

(2) It is difficult to believe that the characters dealt with in the case of Mendelian investigations on animals can be largely affected by environment, but it is easy to allow for this influence by the method of partial correlation. If, in an investigation on any given character, the subscript 1 denote offspring, 2 the ancestor in any generation, 3 the offspring’s environment, and 4 the ancestor’s environment, then the correlation between 1 and 2 for constant 3 and 4 is given by

___________ 7*12(1— 7*342) — 7*137*23—7*147*24 + 7*34 (r14r23 + 7*i3r24)_________34 12 { 1 — 7*232 — 7*34* — 7*24a + 3 r 23/’347*24}* { 1 — TV/ — — + 2 ^ 3^34^ 4}^ '

If we suppose the environment of a given stock to remain constant, this reduces to the simple form

where e is the correlation between character and environment.* This should be absolutely true for pairs of brothers reared in the same home conditions.

Many investigations into the value of e for a great variety of characters in man have been carried out in the Statistical Laboratory at University College, and these show that the value of e is remarkably small. Its average value is 0*03, and it rarely reaches an intensity as high as 0*1. I t can only introduce a second order correction into the correlation, and one well within the limits of the probable error of the determination. If e were 0*1, the change in fraternal correlation would only be from 0*50 to 0*495. Thus, until the number and accuracy of observations have reached a much higher stage, the influence of environment need not be considered.

(3) A noteworthy conclusion reached by biometric workers by actual measurements on man is that the correlation of brothers is not much larger than the correlation of parent and • offspring. A priori, we

* A Germ an w rite r has recently given a form ula for correcting fra ternal correlation for the influence of environm ent. I t differs from the above form, and consists in a factor m ultip ly ing rn. H e assumes th a t the high values found for the paren tal and fra te rn a l correlations in m an as compared w ith those to be expected for somatic characters in the M endelian hypothesis are due to the neglect of the influence of environm ent by the biometricians. H e was probably unaw are of the am ount of atten tion which has been given by the biom etric school to the determ ination of environm ental correlations and to the comparison of the correlations for characters where environm ent is likely and unlikely to produce effect, e.g. hered ity in sta tu re ar.d forearm with heredity in eye colour and cephalic index. See W . W einsberg, i Z eitschrift fu r V ererbungslehre ; 1909, vol. 1.



should expect it to be larger, for brothers have two parents in common, while between father and son comes in the influence of the mother, exerted only on the latter member of the pair. Thus Sir Francis Galton, in his ‘Natural Inheritance’ (p. 133), makes the fraternal resemblance twice as great as the paternal. Pearson*, dealing with three characters in 12 tables of over 1000 each, found the average correlation for parent and child to be 0-46, and for three characters in nine tables the average fraternal correlation to be 052. For eye-colour in mailt he found the parental correlation 0-495 and the fraternal 0475, showing a slightly reversed relationship. Both, however, compel us to affirm that the degrees of resemblance between parent and offspring and between brethren are very nearly equal and differ but slightly from 0-5. The values which Mendelian theory leads us to expect are shown in Table IX.

(4) Further conclusions can be drawn now that the measurement of resemblance between pairs of cousins and between uncles or aunts and nephews or nieces has been completed. Judged by a great variety of characters, the cousin relationship has been found to be as close as, if not closer than, the avuncular relationship. This result is, of course, no more remarkable than that between the parental and fraternal resemblances referred to above. However, in view of the publication of a large number of avuncular correlations which have been worked out in the Statistical Laboratory at University College, it seems desirable to show that it is at least consistent with some one or other theory of heredity. From Table IX it will be seen that on the Mendelian theory the relationship between cousins is as close as that between uncle and nephew.

The point is one which is of considerable importance in medical diagnosis. The parents’ brothers and sisters have been usually included, and the cousins excluded, in considering the family history. The fact, however, that a man’s cousins resemble him as closely as do his uncles and aunts shows that the cousins, usually more numerous, provide at least as good material for an estimate of his tendencies as his parents’ brothers and sisters.

The paradox that a man’s father’s brother’s son resembles him, on the average, as closely as his father’s brother, although the additional influence of the father’s brother’s wife has come in, is only a paradox so long as we overlook the fact that somatic and gametic characters are not causally related but only correlated. Almost any determinantal theory of heredity brings out the fact that, on the average, a man resembles his father as closely as he does his brother, and, further, that he is not more like to his

* ‘ Biometrika,’ vol. 2, pp. 378, 387.t ‘ Phil. Trans.,’ A, 1900, vol. 195, p. 106.

1910.] in a MendelianPopulation Mating at Random. 39



uncle than to his uncle’s son. This point is illustrated in the present paper from the investigation of the collateral correlations of a simple Mendelian population mating at random because that determinantal theory is much in vogue to-day. It will be used here to indicate that the facts reached by observation as to the degrees of resemblance between an individual and his ascendants and collaterals, and considered by some to be paradoxical, are curiously enough legitimate developments of Mendelian theory. If they are paradoxes of biometric observations, they are also paradoxes of Mendelian theory, and highly probably of other determinantal hypotheses which may hereafter be developed.

(5) Tlie three first degrees of collateral relationship are worked out in the following paragraphs, viz.:—

(A) The correlation of siblings, i.e.of brethren regardless of sex.(B) The correlation of uncles and aunts with nephews and nieces.*(C) The correlation of cousins r egardless of sex.

The algebra presents no special difficulties beyond laboriousness.(ti) Starting with the population of the form

p2 (AA) + 2 pq(A + ( ),

after 16sf random matings of pairs of individuals of each class, the types of families produced will be those given in the square brackets in Table I. The frequencies of the families will be the factors which multiply the expressions within the brackets. Thus, for example, the mating of the p2 individuals possessing the protogenic constituent (AA) with the 2 possessing the hybrid (Aa) gives rise to 2 pzqfamilies, in each of which 8s individuals have the protogenic constituent (AA) and 8s the hybrid constituent (Aa).

40 Mr. E. C. Snow. Correlations between Collaterals [June 3,

* The word “ siblings ”has been reintroduced and largely accepted as a convenient term for brethren of both sexes, bu t no like terms are in use to denote uncle or aunt and nephew or niece regardless of sex. Prof. Pearson has suggested for uncle oraunt and sibmag for nephew or niece as convenient Anglo-Saxon names.

t The num ber of matings is taken of the form 16 .s in order to p revent fractions occurring later on in the paper. See Table IV .




Table I.

Secondparent.

F ir st parent.

P (A A ). 2pq (A a ). <i* (« « )•

i>2 (A A) p 4 [1 6 s (A A )]*

2 psq [8 s (A A ) 4 8 s (A a) ] p 2q2 [1 6 s (A a )]

2pq (A a) 2p*q [8 s (A A ) 4 8 s (A a )] 4 p 2q2 [ 4 s (A A ) 4 8 s (A a ) 4 4 s (a a ) ] 2pq3 [ 8 s (A a ) 4 8 s (a a )]

q2 (aa) p 2q2 [1 6 s (A a )] 2pqs [ 8 s (A a ) 4 8 s (a a ) ] q4 [1 6 s (a a ) ]

(A.) Correlation between Brothers.(7) From Table I we can pick out all the pairs of brothers and arrange

them according to the gametic character exhibited by each member of the pair. The cases which can arise, and the number of pairs of brothers in each case are:—

Both brothers (AA)—p4. J . 16s (16s— l) + 2pzq. \ . 8s (8s— 1)

+ 2pzq.| .8 s (8 s — l) + 4Jp222- l . 4s(4s—1).Both brothers (A a)—

2 • 8s (8s— 1) +f<f. | . 16s (16s—1) + . | . 8s (8s— 1)+ 4 P2q2. \. 8 s ( 8 s - 1) + 2 p<f. i . 8s ( 8 s - 1 . i . 16s ( 1 6 s - 1)+ 2i?23. i - 8s(8s—1).

Both brothers (ad)—i . 4s(4s—1) + 2pqs. \. 8s(8s— 1) + 2p<f. ^ . 8s (8s— l)

+ <?4. ^ . 16s (16s—1).One brother (AA), other brother (A a)—

2pzq . S s . 8s+2p3q. 8 s . 8s + 4/>2 2. 4 s . 8s.One brother (AA), other brother (aa)—

4^2 2. 4 s . 4s.< )ne brother (Aa), other brother (aa)—

4 ^ y . 8 s . 4s+ 2/?23. 8 s . 8s+ 2 ^ 3. 8 s . 8s.

Irom the above expressions a symmetrical table can be constructed, from which to obtain the gametic correlation between brothers. This is given in Table II. By arranging the above expressions under the headings “ A ” and ‘‘ not A ” another table can be formed to give the somatic correlation. This is given in Table III. In both tables the factor 16 s, which multiplies every term occurring, has, for brevity, been omitted.




Table II.—Gametic Correlation between Brothers. (Factor 16s omitted throughout.)

Second* First Brother. 1!

brother.(AA.) (Aa.) (aa.)

Totals.

(AA) (16 s - 1) jt?4 + (16 s - 2) p*q + (4i8—l)p?q2

16sp*q + 8 sp2q2 4 sp2q2 ( 1 6 s - l ) ^ 2 (jJ+ g,)

(Aa) 16 sp*q + 8 sp2q2 ( 1 6 s - 2) jfiq + 4 (12 # - 1) + (16 s —2) pq*

8 sp2q2 + 16 spq8 2 (16# —l)jpg

(aa) 4 sp2q2 8 sp2q2 + 16 spq* (4 s — 1) p 2q2 + (16 s—2)pq3 + (16 s —l)^ 4

(16 s — 1) g2 (p + q)'

Total 8 (16 s —l ) p 2 (p + q)2 2 (16s—l) p q (p + q)2 (16 s — 1) q2 ( p + q)2 (16#—1) (p + g)4

Table III .—Somatic Correlation between Brothers. (Factor 16s omitted throughout.)

SecondFirst brother.

Totals.brother.(A.) (Not A.)

(A) (16s—1) p 4 + 4 (16 s—l ) p 3q + (68s—5) p 2q2 + (16s—2)pqn.

12sp2q2 + 16 spq* (16s-l)^>(p + g)2(/» + 2g)

(Not A) 12 sp2q2 + 16sjpj3 (4 s— 1) p2q2 + (16 s—2)pq* + (16s —1) q4

(16 # — 1) g2 + g)2

Totals (16s —1 ) p (jt> + g)2(jp + 2g) (16s — 1) q2 (p + q)2 (16 s —1) (p + g)4

Since the standard deviations of the totals in these tables are the same inboth directions, the correlations can at once be found from a knowledge of the means of the columns.

(8) Gametic Correlation.—If, in Table II, y\, y% y%, and y are the means of the first, second, and third columns, and of the whole table respectively, each being measured from the centre of the middle row, we find

_ (16s—1) jt?2 + (16s — 2) pq—<f (16s—1)»—1 (16s— l)(p+g')a (16s—l)(y>+<?)’

_ 8 s (p2—q2) = 8 s (p -g )2/2 (16s— l)(p + ?)2 ( i 6s—1) ’ 1- - p3- ( 1 6 s - 2 ) ^ - ( 1 6 s - l ) g 2 _ ff—(16s—1 y* ~ (16s—1) ( ? + )2 (16s—l)(y>+2)>

16s(16s— 1) (p + q)2 (y2—q2) g* = io«(i8>-:i)&+gj8~...= p+a'




Hence y \—fa = y j — r,

from which it follows that the regression is linear and the correlation is8 s—1 1 6 s - r

(9) Somatic Correlation.—In this case, from Table III, we have, measuring from the “ not A ” row,

- _ ( 1 0 l ) y 3 + 4(16s— l)jo2yH-(68s—5 ) ^ 2-f (16s — 2)g3 (16s—1)(^> + ?)2 (p + ’2q)

_ _ 12 sp + 16 //2 ~ (1 6 s - l) (p + ?)

_ p ( p + 2q)(p + q f ’

,2’

U \-!h(4s — 1) ;j + (16s—2)

(16s—1)(^ + 2^)r .

The somatic correlation therefore depends upon the ratio of p to q, i.e. upon the ratio of the number of individuals in the population possessing the dominant character to the number possessing the recedent. For different

z|. g_^values of p and q the above value of r can range between and1 lbs—2 ■ 2 0 s -3 *2 • I s i r r 11 p = qits value is

(10) In considering the value to be given to s in these formuhe in order that they can be compared with statistical results, we must remember that the latter are obtained from families of varying size, from which we take selections of fraternal couples. We must therefore look upon our observations as forming a random sample of all the possible pairs of brethren obtained from an indefinitely large number of matings. That is, we must make s indefinitely large. Then we get the gametic correlation to be £, and the range for the somatic correlation from \ to | , with the value when p = q.

* This particular value of the somatic fra ternal correlation was reached by Prof. Pearson in his memoir in the ‘ Phil. Trans.,’ A, 1904, vol. 203. In th a t memoir he dealt with somatic characters only, and he assumed the population to be generated from a series of hybridisations and not from a combination of individuals exhibiting the protogenic, allogenic and hybrid constituents in certain proportions. B ut he was more geneial in his supposition th a t the character depended upon n M endelian couplets, and not on a single one.



(B.) Correlation between Uncle and Nephew.(11) In this case we require not only the various types of families given

in Table I, but also the offspring of these families when mated at random. But it will not be sufficient to mate the nine types of families given by Table I with the general population in which the proportions possessing the protogenic, hybrid, and allogenic constituents are AA), andq2(aa), for any particular mating will he of a member of one type of family with a member of the same or another type. An avuncular relationship will arise with the members of both families from which the mating couples are selected. Thus we must discover the offspring which arise from the mating of each type of family in Table I with itself and with each other. This is given in Table IV, in which 4 tis taken to he the number of matings of each pair. This allows for a possible change of fertility in consecutive generations.

(12) To explain Table IV, consider, for example, the mating of the family of type 4s(AA) + 8s(A«) + 4s(aft) [Type a ] with the family of type 8s(AA) + 8s(Aft) [Type /3]. Of the 8s (AA) of Type /3 :—

2s will mate with 2s of the 4s(AA) from Type a, giving 4£{2s(AA)} offspring.

4s will mate with 4s of the 8s (Aft) from Type «, giving ‘I t {4s(AA) } + 2 t{4s(Aft) } offspring.

2s will mace with 2s of the 4s(ftft) from Type a, giving (Aft) }offspring.

Of the 8s (Aft) of Type /3 :—2s will mate with 2s of the 4s(AA) from Type «, giving '2t {2s (AA) } -f

2<{2s(A«)} offspring.4s will mate with 4s of the 8s (Aft) from Type a, giving £{4s(AA)} +

2t{4s(Aft) } +t{4s(ftft) } offspring.2s will mate with 2s of the 4s(ftft) from Type a, giving 2 {2s(Aft) } +

2 t{2s(ftft) } offspring.

The complete result will consist of the aggregation of the six combinations written out. Since the frequency of the Type a family is and of theType /3 is 4 p3q,there will be altogether 1 such aggregations as the above. The result is represented in the table, in which the families derived through any particular individual exhibiting the constituent (AA), (Aft), or (ftft) are written in the same horizontal or vertical line as the constituent itself. The remaining aggregations can be read off in like manner.

(13) From Table IV we can pick out uncle and nephew in pairs according




Snow. Roy. Soc. Proc., B . vol. 83, to face page 44.

Table IY.—Showing the Types and Frequencies of the Families of the Second Generation.----------- -

p 4 4p3 4p2q2 2 p Y(__ -A- _]4 pq3

A A

' 16s (A A )— -------*-------------------------'

85 (AA) + 85 (A a)__ _A_ __ ^

45 (A A ) + 85 (A a ) + 45 (aa) I 65 (A a)t ^

8 5 (A a) + 85 (aa)t ^

165 (aa)

Ps 4p^q 4pQq2 2 p 6q2 4sphqz pY

p 4^ 16.9 (AA)1----------- n4£ [16 s (A A )] ' A*TQHi AM A. 2 * [8 s (A A ) ]4st[8 s (A A )] + + 2 ^ 8 s (A «)J ’ « [4* (A A )] + ♦ « E l : ( A ^ / 4 4 ( [ 4 . (A .) ]

1 2 t [ 1 6 5 (A A )]1 + 2 £ [1 6 5 (A a )] 1 ' ♦ « E £ E $ i l+ 4 * [ s . < A .)]'

1 1 4t [1 6 s (A a )]

4p7 16p6g2 16pbq* 8p5q3 16 >4 4 4 p sq&

8 s (AA)r n

4£ [85 (A A )] ' 4 < [ * . ( A A » ♦ $ # £ $ 4 t [ 2 s (A A )] + + 2 ^ i s ( A a ) ] 1 + 4 ^ 2 " (A^2 £ [85 (A A )] ;

+ 2 £ [8 5 (A a )]2 ^ [4 s (A A )] 4 t[ 4 s (A.a)~\+ 2t[4 s (A a )] + 4 ^L4 -HA «)J

I ................... 14^ [85 (A a)]

4p3q« +

8 (Aa)

+2 * [8 s (A A ) ]

+ 2t [ 8 5 (A a )] + 2^ [4 s (A a )] + ^ 4,5 (a a )]

+ + +

2 # [ 2 s (A A )] +2tS t s ( i t ) l 2 # [2 s (A a)]+ 2 t [25 (A a)j + j4(y (a a )] + 2 * [2 5 (a a )]

+£ [8 5 (A A )]

+ 2 t [85 (A a)]+ £ [8 5 (a a ) ] |

+ +^ [4 5 (A A )] 2 ^r4 5rA a 1

+ 2# [4 s (A a )] f+ t [4 s (a a )] + 2 s (a a )]

+2 1 [85 (A a)]

+ 2 t [85 (a a )]

4p6q2 16jp5 3 IQ/?4#4 8 p4q4 16p'"qb 4 i ? y

45 (AA) 4^ [45 (A A )] . , ro / n n , 2 £ [ 2 s ( A A ) ] 4^ [2 s (A A )] + + 2 ^ 2 s ( A a ) l '4 < [ ' ( A A ) ] + + | ( E | ; ( i ^ ? + 4 i [ . ( A » ) ]

2 £ [4 5 (A A )] 1 + 2 £ [ 4 5 (A a)] j 2 [2 s ( A A )] 4 /r2s('A a ')1

+ 2t[2 s (A a )] + 4 G ^ * [A a ; j |1 1

4 ^[4 5 (A a)]

+ + + + + 4- + + + + +

4p2q2 -

• 1

<5 2# [8 s (A A )]

+ 2 £ [85 (A a)]9 / ra«CAAM # [4 s ( A A )]fG 4 s (A A )J + [4 s (A a )]

+ 2 * [4 s (A a )] + [ 4a. (a a ) l2 < [2 .(A A )[ | . „ ' f t “ !A A )] 2 / [2 s (A a )]

+ 22 L2s (Art)] + 2< P 4 M 3

[85 (A A )] i + 2 t [85 (A a )] | + ^ [8 5 (a a )]

+ « E £ E ^ ) 9 j ' F i ' C - j j+ t [ 4 s ( a a ) ] + 2 ^ [ 4 s ( a a ) ]

2 [85 (A a )] + 2 1 [85 (aa )]

+ + + + + + + + + + +

45 (aa) \ t [45 (A a )]. , r„ /A N-, 2 £ [ 2 s ( A a ) ]4£ [2 s (A a)] + 2 [ 2 s (a a )] 4< [«< A »)] + ! 5 g * . E i ^ J 4 , [*<"*>]

2 ^ [4 5 (A a)] + 2 t [45 (a a )]

2 ^ [2 5 (A a )] 4 f2s (aaW+ 2« [ 2 s (a a ) ] 4 « L J s [a a )j 4^ [45 (a a )]

2p6q2 8pbq3 V ? 4 4 p4q4 8p3qb 2 p Y

2jpV j 16s (Aa)

r ------------12 t [ I 65 (A A )]

+ 2t [165 (A a )]o /rS fC A A -)! £ [ 8 s (A A )]

J T8 s Aa?i + + 2 * [S s (A a ) ]+ 2 r [8 s (Aa)J + ^ 8 s (a a )]

p 12£ (4 s (A A )] . 2 4 f s ^ A a ? i ] + 2 G 4 ^ (A a )]

+ 2 d (4 s (A a)] + t ; C[ 8 s (a a )] + 2 * [4 * (<**)]

^ [1 6 5 (A A )] + 2 t [ I 6 5 (Aa)] + ^ [1 6 5 (a a )]

+ 2 / r M ( ! i l 4- 2^ [ 8 s (A a)]+ 2t[8 s (A a ) ] + o . >+ [ 8 s (a a )] + 2 f [ 8 s ( a a ) J

i i2 # [1 6 s (A a )]

+ 2 £ [16s (a a )]

4pbq3 16jp4q4 16p325 8p3qh 16 p Y 4jt?^7

1r

8 5 (Aa)

r n2t [85 (A A )]

+ 2t [85 (A a )] ' ‘ ‘ f i ' q A f l+ 2 1 [4 5 (Aa) ] t[4 s (A A )] ]4 s (A a)] 45 (a a )]

2 * [2 s (A A )] „ J f J * . 2^ [2 s (A a)]♦ « t * . W r t ’ l E i l E i S ? 4 « t » . ( ~ ) ]

r t[8 s (A A )j , + 2# [8 s (A a )] ; + ^ [8 5 (a a )] + t[4 s (a a )] + 2 < [4 s (a a ) ]

1-------- n2 [85 (A a)]

+ 2 / [85 (a a )]

4pq3* + + + + + + + + + + +

85 (aa) 4 t [85 (A a )] 4 ( [ 2 . ( A » ) ] « C « 4 ( ~ ) ]2^ [8 s (A a)]

+ 2t[8 s (aa )]2^ [45 (A a )] Aj.rA f m

+ 2 * [ 4 s ( « a )J 4^ [4 s (a a )] 4s t [85 (a a )]

pY 4p3qb 4/?2<76 2 pY 4 pq7 !?8

q4 * |l65 (aa)r n4 t [16 5 (A a )]

r , , r„ ,. 2 £ [ 8 s ( A a ) ]4* [8 s (A a)] + + 2< [8 / (aa)] 1 4 ( [ 4 s (A a)] + + |'E S iE iS ? + 4 ,[1 *(““)]1 2t[1 6 s (A a )]1 + 2 ^ [1 6 5 (a a )] |

2 # [8 s ( A a ) ] , . ra . .J1+ 2 / [ 8 « ( a a ) J + 4 ^ [ 8 s ( a a ) ]

r n4^ [1 65 (aa )]



to the particular constituent, protogenic, hybrid, or allogenic, each member of the pair possesses. I t will be more convenient, however, to make use of the fact that the number of pairs of uncle and nepbew will be equal to the number of pairs obtained by taking anv one of the childien fiom given parents with any one of the grandchildren, minus the number of pairs of parents and offspring among such pairs. This latter can be obtained more simply than by Table IV. Tor Table I enables us to pick out parent and offspring in pairs and arrange them according as each member of the pair is (AA), (Ad), or (da). But we require the number of such pairs in the next generation. Allowing for the change in the fertility from 16 s to 4£, it is clear that the number in the second generation will be obtained from the number in the first by writing for 16s and multiplying by 16s ( )4. FromTable I we can at once pick out the following terms for the first generation.


itParent (AA), offspring (AA)----.. p4. 18s + 2p3q.8s,

„ (AA), 9) (A a)..... 2p3q. 8 s + p 2q2.16 s,„ (AA), 1? (aa) __ .. o,„ (Aa), (A A).... 2 p3q. 8s + 4 ppq2. 4s,„ (Aa), V (A a).... 2p3q. 8 s -j- 4=p2q2.8 s + . 8 s,,, (Aa), }> (aa) __ .. 4p2q2.4:S+2pq3.8s,„ (aa), » (AA).... .. 0,„ (aa), yy (A a)__ .. p 2q2,16s+2pq3,8s,„ (aa), yy (aa) __ 2pq3.8s + <?4.16s.

In obtaining these expressions, one parent only was considered. From the symmetry of the table, if both parents are to be taken into account, each term must be multiplied by two. Since this factor will occur throughout the work, it can be omitted.

Collecting the above terms, and applying the modification necessary to transform the first generation into the second, we have, for this lattergeneration, the frequencies :—

Parent (AA), offspring (AA)....... (^+<?)5,„ (AA), „ (A a)........„ (AA), „ (aa) ........ 0,„ (Aa), „ (AA)........„ (Aa), „ (A a)........ 64„ (Aa), „ (aa) ....... 64 (p q)b,,, (aa), „ (A A)........ 0,» (a»), „ (A a)........ 64stpq* (p + qf,

„(aa), „ (aa) ........ .64From Table IV we have now to pick out and arrange the various pairs



which can be obtained by taking any one in the first generation with any one of the same stock in the second generation both sons and nephews).

To economise space, the details of the terms are given in the case of three only of the nine pairs. The other six can be written out in the same manner, but the results alone are stated. The three pairs given in detail are:—

Member of the first generation (AA), member of the second generation (AA)—

p s. 16s.4£. 16s + 4 p7q. 16s. (4 t. 8s + 2£. 8s)+ 4 p6q2. 16s (4 t. 4s-f 2 t .8s)+ 2 p6q. 16s .2 16s+ 4 p hqz. 16s. 2 t. 8s + 4zp"lq. 8s(4£. 8s + 2£. 8s)+ 16 p6q2. 8s ( 4t . 4:S+2t. 4 s + 2 t. 4s + t . 4s)+1 6pbqz . 8s (4£. 2s + 2 t . 4s + 2£. 2s + 1 .4s)+ 8 pb<f. 8s {2t. 8s+ t. 8s) + 16y>V. 8s(2 . 4s+ . 4s)

+4:p*q2 A s{k t As+2t. 8 s ) + 1 6 ^ 3. 4s 2 s+ 2 t . 4 s+ 2£ . 2s+ £ . 4s)+ 16j?4g4. 4s (4£. s + 2 t . 2s + 2 t . 2s + . 4s) + 8 p^q4. 4s (2 t. 4s + t. 8s) + l§pzqz. 4s (2 . 2s+ . 4s).

When these terms are collected, and simplification performed, the whole expression reduces to the simple form

512s2tp A (p -\-qY('2p +Member of first generation (AA), member of second generation (Aa)—

4 p7q. 16s.2 t .8s + 4 pzq2. 16s(2 t .8s + 4£. 4s)+ . 16s. 2 16s + 4 p \ z . 16s (2 t. 8s + 4£. 8s)+;p424- 16s. ^ • 16s + 4 pq. 8s . 2 t .8 s+ 16 p6q2. 8s (2 t. 4s+ 2 . 4s + 2i . 4s)+1 6pbqz . 8s (2 t .2s + 2t,.4s + 2 t .4s + 4f . 2s + 2s)+ 8pb( f . 8s (21. 8s+ 2 t . 8s)+ \Qp/*q* . 8 s (2 t . 4s+ . 4s + 4A 4s+ 2 . 4s)+ 4 pzq5. 8s(4 t. 8s + 2 t . 8s) + 4jJ»(i 2. 4s(2£. 8s + 4£. 4s)+ 16pV . 4s (2 t. 4s + 4 t. 2s-\-2t. 2s+ . 4s+ . 2s)+ 16pV .4s(2+ 2s+ 4* .s+ 2 t. 2s+2*.4s + 2*.2s + 4*.s+2*. 2s)+ 8 pY. 4s (2 t. 4s + 2t.8s+ 2 t.4s)+1 (jpY. 4s (2 t. 2s + 2*. 4s+ 2t. 2s + 4*. 2s+ .4s)+ 4 p2(f. 4s (4/. 4s + 2 t .8s).

This, when simplified, becomes512 sHp>2q(p + q f (op + q).




47

Member of first generation (AA), member of second generation\§p*q2. 8 s . t .4s + 1 5 p \z .8s (t .4s + + 8 .8 s . t .8 s

+ Wpiqi . 8s (t. ks + 2 t . 4s)-f 4p3#5. 8s. 8s+ 1 6/rr,§3.4s (t. 4.9 4- 2 t . 2 s) + 1 6piq4: . 4s (tf. 4s 4" 2 £. 2s-\-2t. 2 s+ 4 £. s)+ 8 pY■ 4 s(t.8s+ 2£ .4s)+ 16^325. 4s (« A s + 2 t. 2s4s + 4£. 2s) + 4 p2q6 . 4s(2£. 8s4-4£ .4s).

This reduces to 512s2tp 2q2 (p + q)*.

The frequencies of the other pairs are (the member of the first generation in each case being placed first)

(Aa) with (AA)....... 512 s2tp 2q ( 4- )4 4-(Aa) „ (A a)....... 512 $*tpq(Aa) „(aa) 512 s*tpq2 (p + qY (p + Sq),(aa) „(A A) 512s?tp2q(aa) „(A a ) . 512 s2tpq2 + qY + 3^),(aa) „ (aa) ........ 512s2tq3 (

1910.] in a MendelianPopulation Mating at Random.

Subtracting from these expressions those found for the corresponding combinations in the case of parent and offspring of the second generation given above, we get the following frequencies for combinations of uncle and nephew:—

Uncle (AA), nephew (AA)...,)> (AA), y y (A a)...y y (AA), » (aa) ...y y (Aa), (AA)...y y (Aa), y y (Aa) ...

y y (Aa), y y (aa) ...(aa), y y (AA)...

y y (aa), y y (Aa) ...>5 (aa), y y (aa) ...

6 4 s^ 3(jp + )4 [(16s— l)p + (8s— 1) q\64 stpq(p + qY [(24s— l)j92 + (8s—1)2*7], 512 s*tp2q2(p + qY.64 stpq (p + qY [(24s—l)p2+ (8s— l)pg[\,64 stpq (p 4- qY[(8 s — 1 )p24- (48 s — 2)pq +

(8s—1) ( f \64 stpq2(p + qY [(8s—1)jp4-(24s—l)#],512 s*tp2q2(p 4 qY,64s£ p>q2 (p4- qy[(8s— l)p4- (24s—1) q\,64 st qz (p4- qY[(8s— 1) ? 4- (16s — 1) q\

Correlation tables can now be formed from which to derive the gametic and the somatic relationships between uncle and nephew. Table V gives the (jametic relationship; Table VI the somatic. In both of these tables the factor 64s£(p4-^)4, which multiplies every cell, has been omitted.



Table V.—Gametic Correlation between Uncle and Nephew.(Factor 64 st (p-f g)*omitted throughout.)


Nephew.Uncle.

----------------------

(AA.) (A a.) (aa .)Totals.

(AA) (16s—l)/>4 + (8s— 1 )p*q[ (24 s — 1 )p*q + (8 s—l)p 2q2 8 sp2q%i (16s—1) >2 G> + g)5|(A®) (24s - 1 ) p \ + (8 s - 1 ) p 2q2 (8 s - 1 ) p3? + (48 s - 2) p 2q2

+ (8 s —1) pq3(8 s—l)p \* + (24s — 1) pq* 2 (16 s—1) pq (j» + f)|

(aa) 8sp2q2 (8 —sT)p2q2 + (24s — 1) pq* (8s—1) pq* + (16s—1) qA (16s—1) (p + qy \

Totals (16s—1) p2 (p + q)2 2(16s—1) pq( p + Sy (16*—1) 2* (j»‘+j)* * (16s—1) ( i

Table VI.—Somatic Correlation between Uncle and Nephew. (Factor 64 st(p + qYomitted throughout.)

fVj prvn our

Uncle.Totals.li cuiic n •

(A.) (Not A.)

(A) (16s—l ) p 4 + 4(16s—1) p*q + 4 (16s—l)p 2g2 + (8s— l ) p q z

(16s—1) p-q2+ (24s—1) 3(16s-l)i> (p + q)2 + 2q)

(Not A) (16s—l ) p 2q2 + (24s—l)pq* (8s—l)i>23 + (1 6 s —1) 4 (16s —1) q2 (p + q)2

Totals ( 1 6 s - 1) p {p + q)2 (p + 2q) (16s—1) q2 (p 4* q)2 (16s—1) (p + g)4

(14) Gametic Correlation.—From Table V we find, measuring from the centre of the middle row—

_ (16s—l)/>2 + (8s—1 —Jl (1 6 s - l ) (p + 2)2

y * = t

Hence

_ 8 sp2—(8s—1 —(16s—l)? 2W (1 6 * - l) (p + 2)> :

l ) (p 2- q 2) _ p*-q3 _p q- l ) { p + q f {p+q)2 p + q

_ _ 8s—1 1 _ -y '-y * = • g = 2/2—2/3 = r.

Therefore the regression is linear and the correlation 8 s - 1 As before,16s—1 ’ 2 'we must take a random sample from the families obtained by making s indefinitely large. The correlation then becomes



1910.] in a Mendelian Population Mating at Random.49

(15) Somatic Correlation.—Table'VI gives, measuring from the “ not A ” row—

= p(p±2q)J o + ? ) 2

and —2/2 8 s - 1 _ _ J [ _ _16s— 1 ’ p +

Again the correlation depends upon the relative values of and The limits between which rcan lie for different values of and q are 0 and

1^~~"T ^ P == 7’ va ue 4S iQg" \ • w • Putting s infinite, the range

for r is from 0 to }, with the value | when

(C.) Correlation between First Cousins.

(16) From Table IV we can pick out every pair of first cousins, and arrange them in groups according as each member of the pair exhibits the constituent (AA), (A a)or (aa). But, again, it is more convenient to obtainthese groups indirectly. For the number of pairs of cousins in any group will be equal to the number of pairs which can be taken from all children having the same two grandparents, minus the number of pairs of brothers among such children. Since the population is stable during succeeding generations, the latter can be obtained at once from the expressions given in (7) by writing 4:t for 16s in that table and multiplying by the factor 16s(p + 2’)4. We therefore have the following expressions for the number of pairs of brothers among all grandchildren :—

F irst Secondbrother. brother.(AA) . (AA).(AA) (Aa) .(AA) (aa) .(A a) (Aa) .(A a) (aa) .(aa) (aa) .

32st (p + q)4[(4 t-l)p 4 + (4* - 2)p2q + ( t - l )64st (p + qY [4 tp zq-\-2t p2q2~\,64s£ {p + qY t p2q2,32st (p + qY [(4 t- 2)phq + 4 (31 1)p2q2+(4* - 2)pq2], 64 st{p + cqY\2t p2q2 At pq2],S2st (p-\-qY [(t- 1)p 2q2 + (4 t + (4*-1 ) 4].

The number of pairs of all possible grandchildren ( siblings, together with cousins) can be easily obtained from inspection of Table IV. From symmetry, only six combinations are now required. Three of these are given in detail, the final expressions only of the other three being written down. Throughout the detailed expressions k is written for the product s.t.

VOL. lxxxiii.— B. E



First grandchild (AA), second grandchild (AA)—f± . 64 7 (6 4 7 -1 )+ 4 tfq.1 .4 8k(4 8 7 -1 )+ 4 . . 32 (32A - 1)

+ 2p6q2 . | . 327(327—l) + 4p523 . -§•. 167 (167—1) + 4Y? . | . 487 (487 — 1)+ 1 Qp6q2. \ . 367(367—1)4- 16p5 3. \ . 247(247—1)4- 8Y#3 . | . 247(247— 1)

4-16 j?V . i . 127; (127-1)■4- ± p \2. i . 32 7 (3 2 7 -1 ) 4- 16Y<?3. | . 247; (247c-1)

4- 16jpY* i . 167;(167; —1)4-8j?4</4. £ . 167;(167; —l) + 16jpY- . 87;(87;—1)+ 2 peq2. 4.32 h(327—1) + 8jY . i . 247c (247;—1) + 8 ^ Y . \ . 167c (167'— 1)

4-ApY-i* 167(167—1) + 8Y<?5- i .87 (87—1)4-4 p Y .4 .1 6 7 (167—1)+1 6pV .|.1 2 7 (1 2 7 -1 )4 -1 6 j? Y . £ . 87 (87— 1) + . § . 8 7 (87—1)

+ 16p2q6.% .4 7 (4 7 -1 ) .This reduces to

32 stp2(p + q f [(64s/ - 1 ) p2 + (32. sZ- 2 + (4 - 1 ) $2].First grandchild (AA), second grandchild (Aa)—

4 p1q. 4 8 7 .167 + 4j»Y • 527c. 327:4-2 . 327 .327c

4-4p523 • 167.487 + 4^2 .487 .167 + I6Y22 • 367.247 + 16^V -247.327+8Y 23. 2 4 7 .3 2 7 + 1 6 p Y -127.407 + 4 pQq2. 327 .327+ 1 6p5qs. 247.327+16+Y . 167.327+ 8 p Y . 1 6 7 .3 2 7 + 1 6 /Y • 8 7 .327 + 2+Y • 327.327 + 8 p Y . 247.327 + 8pY • 167.327 + 4pY • 167.327 + 8 ^Y • 8 7 .3 2 7 + 4 ^ Y -167.487 + 1 6 ^ Y -127.407

+ I6Y25 • 87 .327+ 8Y25• 87 .3 2 7 + I6Y26 • 47-. 247.This becomes 512s2/2 Y? Y + (lY (12j> 4- 3q).First grandchild (AA), second grandchild (aa)—

lQp^q2. 367 .47+ 1 Qp q*. 247.87 + S+'Y • 247.87

+ 16pY • 127.127+16Y23 • 247.87 + 16/;7/1.1 6 7 .167 + 8pY • 167.167+ 16jpY • 87.24/.+ 8 . 247 .87 + 8Y 24 • 167.167 + 4j^q1.167.167

+ 8Y 26 • 8 7 . 2 4 7 + 16+Y • 127.127 + I 6Y 25 • 87.247

+ 8 p3q5. 87 .247 + I 6Y 26 • 47.367.This becomes ■ 2304 s2t2p2q2 (p + q)*The other combinations reduce to :—First grandchild (Aa), second grandchild (Aa)

64 s/ pq (p + qY[(16 s/ - 1 ) P2 + (104s/ - 2) + (16 s/ - 1 )




First grandchild (A a),second grandchild (aa)—512 s2*2 pq2 (p + q)4 (3p +12 §).

First grandchild (aa), second grandchild (aa)—32s* q2 (p+ q)4 [(4s*- 1 ) p2 + (32s*- 2) + (64s*-1) q2].

To obtain the frequencies of the various pairs of cousins, we have to subtract from the above six expressions the corresponding expressions for pairs of brothers. We then get the following frequencies :—

Both cousins (AA)— 32 st2p 2(p+ q)4 [64s—4 ) ^ + (32 s— l ) j 2].

One cousin (AA), other (Aa)... 128st2 p„ (AA), „ (aa) ... 64st2p

Both cousins (Aa)—

128s*2pq (p + q)4 [(8s— 1)p2+ (52s— + ( 8 s - 1) q2].One cousin (Aa), other (aa)—

128 st2pq2(p + q)4 [(12s— 1) p + (48s— 2) q\.

Both cousins (aa)—

32s*2 q2 (p + q)4 [ (4 s - l ) f + (32s—4) pq + (6 4 s-4 ) q2].The correlations tables for cousins can now be formed. Table Y II gives

the arrangement for the gametic relationship, Table V III for the somatic. In both cases the factor 64 st2(p + qY has been omitted. The tables must, of course, be made symmetrical.

1910.] in a Mendelian Population Mating at Random. . 51

Table VII.—Gametic Correlation between Cousins.(Factor 64 st2(p+q)4omitted.)

[SecondFirst cousin.

Totals.[cousin.(AA). (Aa). (aa).

• (AA) (64s — 4)^>4 + (32s—4)®3g + (4s—l)_p222

(96 5 — 4) p3q + (24 5 — 2) p 2q2 (365—l)i?222 4 (1 65 -l)i> 2 (i>4^)2

[ (-MI

(96s-4X p32 + ( 2 4 s -2 )p V (325—4) pzq 4 (2085-12) p 2q2

4 (325—4) pq6

(245—2)p2q2 + (965—4i)pq* 8 (165—V)pq (p + q)'2

I {aa) (36s—1)j >222 (245—2)p2q2 + (965—4)jpj3 (4 5—l ) p 2q2 4 (32 5—4) pq3 4 (645—4) qA

4 (I65—I) q2 (p + q)a

[totals

1* 4 (165—l)jp2 (p + )2 8 (I65 — l )p q (p 4 q)2 4 (165— 1) q2 (p 4 q)2 4 (16 s—1) (jP + ?)4

E 2



Table VIII.—Somatic Correlation between Cousins.(Factor 64 st2(p + qYomitted.)


SecondFirst cousin.

Totals.cousin.(A). (Not A).

(A) 4 (I65 —l)jp4 + 16 (165 — 1)p*q + (2605— l7)_p222 + 4 (85 — 1)pq*

4 (24 s—1) pq* + 3 ( 2 0 s - 1)4 (16s— 1) {p + qYp {p + 2q)

(Not A) 4 (245—l )p q z + 3 (205— 4 (16s—1) 54 + 4 (8 s—l)p 2 3+ (4s-l)pV

4 (16s—1) <p + q)2

Totals 4(16s—1) (p + q)2p ( p + 2q) 4 ( 1 6 s - l ) 22(p + ?)2 4 (16s —1) (

(17) Gametic Correlation.—From Table VII we find, after simplification—- _ (16s—V)p — 8sq- _ (16s—1)Vl ( lQ s - l) (p + q) 2/3 “ (16s—1) (p + q)

( 2 4 s - l ) ( p -q ) _ p - q2(16s— ~C{p + q)

8s —1From these, y i—y 2

is linear, and the correlation

2(16s—1)8s—116s —1 ' 2 ’

p + q

y2—y 3 = r. Hence the regression again

"When s is indefinitely large this

becomes \.(18) Somatic Correlation.—In this case, after reduction, we find—

„ - , -P (p + 2 i ) an(i a , - a , - ( 4 * - i ) p + 4 ( 8 « - i ) j _y -(2, + i f a n d / V 4(16S- 1 ) ( j> + 2 r-

Hence in the case of cousins also the somatic correlation depends upon the ratio of p to q. The range of r for different values of p and q extends from

4s—1 8s—1 If p = qits value is 36s—5 12 (16s—1)’

When s is inde-4(16s—1) 2 (1 6 s - l) 'finitely large, the range is from TJ-g to with the value T3¥ when p = q.

(19) In Table IX, all the correlations which have been worked out from the Mendelian hypothesis of “ unit characters ” are collected together. The values of the parental and grandparental correlations are from Prof. Pearson’s paper.* The rest have been obtained in the present paper.

* “ On the Ancestral Correlations of a Mendelian Population,” ‘ Roy. Soc. Proc.,’ B, 1909, vol. 81.



1910.] in a Mendelian Population Mating at Random. 53

Table IX —List of Mendelian Correlations between an Individual andhis Relatives.

*G-ametic

Somatic correlation.

Relative. correlation.G-eneral. Range. p = q-

P„ wari f 1 g o - l i2 p + 2 q 2 3

n H "no vpn t .1 q i 0—1 14 p + 2q 2 4 6

Brother or sister....... 12

p + 4tq 1 P + 2q 4

1 1 4 2

512

TTnolp or mint 1 g 1 o - l 14 p + 2q 2 4 6

Oonsin ............... 1 p + Sq 1 1 1 34 p + 2q 16 16 4 16

From this table we notice that, on the Mendelian hypothesis, the parental and sibling gametic correlations are equal, while the grandparental, avuncular, and cousin gametic correlations are also equal and are just one-half of the former. In Table X the ancestral and collateral correlations in man and certain animals founded on biometric research, together with the corresponding gametic correlations found above on the Mendelian assumption of “ unit- characters,” are collected.

(20) The following general conclusions can be drawn from the correlations for somatic characters given in Table IX.

(i) If the cousin correlation is greater than the parental. This seemsremarkable, but it must be borne in mind that the condition is derived from the Mendelian idea of “ unit-characters,” and no prediction can be made as to when the condition is fulfilled. The only cousin correlations worked out for the biometric school are those by Miss Elderton. No correlation for a single character in cousins was found to exceed the parental one. In eye-colour alone did the former approach the letter. The parental correlation in this case was 0*495, while the cousin correlation had the values 0*44, 048, and 038 for male and male, male and female, and female and female respectively, the mean value being 0*434.

(ii) If q is small, i.e. if the recedent constituent is rare, as in the case of albinism in man, the correlations founded on Mendelism have low values, and in certain cases approach zero. For pathological characters the figures found by Miss Elderton were certainly much smaller than for other characters in




Table X.—Mean Ancestral and Collateral Correlations in Man and Animals.*

— Parent. Grandparent. Sibling. Uncle. Cousin.

Man ................................................... 0-470 0-317 0-521 0-265 0 -267

Horse................................................... 0-522 0-296

Basset hound ................................... 0-526 0-220 0-508

Greyhound ...................................... 0-532 0-332 0-559

Shorthorn ........................................... 0 -444 0-200 0-530

Mean of biometric results ............... 0-498 0-273 0-5291

j 0 -265 0-267

Gametic correlation on Mendelian 0-500 0-250 0-500 0*250 0-250hypothesis.

1

cousins, but this was ascribed to the incompleteness of the records. No general tendency in the direction of lower correlations for certain characters has yet been noticed in biometric investigations.

(iii) The correlation between siblings is always greater than that between parent and offspring. This is in accordance with a priori expectation (see para. 3 above) and agrees with statistical results.

(iv) The grandparental correlation is always the same as the avuncular. This also agrees fairly well with statistical conclusions.

In his paper of last year Prof. Pearson drew attention to the fact that the

* Each biometric correlation given in the table is the mean of a num ber of values and is compiled from the following sources :—

(1) K arl Pearson and Alice Lee, “ On the Laws of Inheritance in M an,” P arts I andI I , ‘ Biom etrika,’ vols. 2 and 3.

(2) K arl Pearson, “ On Inheritance of the M ental and Moral Characters in Man, ’ibid., vol. 3.

(3) Amy Barrington, Alice Lee and K arl Pearson, “ On Inheritance of Coat Colour inGreyhounds,” ibid., vol. 3.

(4) Amy B arrington and K arl Pearson, “ O n Inheritance of Coat Colour in Cattle,”ibid.,vol. 4.

(5) E thel M. E lderton, “ On the Measure of the Resemblance of F irst Cousins,”‘ Eugenics Laboratory Memoirs,’ IV , 1907.

The characters correlated for man include stature, span, forearm and eye-colour. The sibling correlations include also hair-colour, together w ith those of no less than 16 characters obtained from school records. The mean fraternal correlation based on this homogeneous m aterial was 0'519. The cousin correlations were deduced from eye-coloiu, hair-colour, health, ability, etc. Among the animals, coat-colour was the charactei correlated throughout.



values of the ancestral gametic correlations in a Mendelian population obtained from theoretical considerations were in close accordance with those determined for the somatic characters in biometric researches, while the somatic correlation on the Mendelian hypothesis appeared to be too low, if the principle of absolute dominance was maintained. The results of the present paper for the sibling, avuncular and cousin correlations agree equally well with the statement made for ancestral correlations. The general

somatic correlation for cousins, viz., is always greater than the1o > + 2<2general somatic grandparental and avuncular correlations. The conclusion of Miss Elderton that “ cousins must be classed as equally important with uncles and aunts, and they may eventually turn out to be as important as grandparents,” paradoxical as it appears, is supported by the theory of Mendelism as wTell as by biometric experience. Without making any dogmatic assertion, it seems safe to conclude, both from theory and practice, that the cousin ranks in no way behind aunt or uncle and grandparent in family resemblance. The importance of this deduction in medical diagnosis has already been indicated.

In conclusion I have to thank Prof. Pearson for considerable advice while the work in connection with this paper has been in progress.

1910.] in a Mendelian Population Mating at Random. 55



Documents

Simple MendelianPopulation Mating at Random. 37rspb.royalsocietypublishing.org/content/royprsb/83/561/37.full.pdf · See W. Weinsberg, i Zeitschrift fur Vererbungslehre ; 1909, vol