September 29, 1904] 



NATURE 



529 



LETTERS TO THE EDITOR. 



\Thc Editor does not hold himself responsible for opinions 



expressed by his correspondents. Neither can he undertake 



to return, or to correspond with the writers of, rejected 



manuscripts intended for this or any other part of Nature. 



No notice is taken of anonymous communications.] 

 Average Number of Kinsfolk in each Degree. 



What is the averag"e number of brothers, sisters, uncles, 

 nephews, nieces, first cousins, &c., that each person 

 possesses? I had occasion to compute this for a particular 

 collection of persons ; the results were so far unexpected as to 

 show that the question deserved a consideration which it has 

 not yet received, so far as I am aware. The problem proved 

 easy enough in the end, but not at first, for there are other 

 ways of attacking it, in which I blundered and lost time. 



The simplest conditions that will serve for a general 

 theory are those of a supposed population (i) the numbers 

 of which are statistically constant in successive generations ; 

 (2) the generations of w-hich do not overlap; and (3) which 

 are "completed" by having wholly passed into history; 

 and again (4) where every person is taken into account, at 

 whatever age he or she may have died. It will be a further 

 great simplification if it be allowed (5) to suppose the males 

 and females to be equal in number, and in all respects to 

 admit of similar statistical treatment. This need be only 

 a provisional way of looking at the problem, for it will be 

 seen that corrections can easily be introduced if desired. 



It will much facilitate matters to begin by dealing 

 exclusively with either the male or the female half of the 

 population, leaving the other half to follow suit. We will 

 begin with the females. 



Let d be the average number of female children born of 

 each woman who is a mother, so if there be n mothers in 

 the population the total number of females in the next 

 generation will be nd. How many of these latter will prove 

 fertile of female children ? On the supposition of statistical 

 constancy, the number of mothers in the two generations 

 will be the same, therefore d out of the nd will be fertile of 

 female children ; conversely, the probability that any one 

 of these female children will herself bear one or more 

 female children =i!d. As a test of this, the average 

 number of fertile daughters to each mother will be 

 </xi/d=i, as it should be. 



Ne.xt, as regards sisterhoods. Each mother bears on the 

 average d female and d male children, or 2d individuals in 

 all. Each of these will have zd—i brothers and sisters, 

 and half that number of sisters, namely, d — J. 



The syllable si will be used to e.xpress " sisters " with- 

 out regard to age or fertility, and si' to e.xpress " sisters 

 who are fertile of female children " ; similarly da and da' for 

 daughters. 



The number therefore of si is d — \, of si' it is (d — ^)/i/, of 

 da it is d, of da' it is I. The number of me', or of mothers 

 10 a child, is, of course, i, and there is no occasion for using 

 iiif, as a mother must be fertile. 



.\ few examples of results are given in the following 

 table ; it could have been extended indefinitely, but these 

 are quite sufficient for drawing conclusions : — 



The foregoing remarks and table are equally applicable to 

 males if bro (brother) is substituted for si, son for da, 

 fa (father) for me. 



It will, then, be understood that each mother, father, 

 or fertile couple has, on the average, d sons and d 

 daughters, or 2d children altogether, of whom i is a fertile 

 son, I a fertile daughter, and that the others die without 

 issue. In the collection mentioned above, the value of d 

 was about 25, that is to say, an average family consisted 

 of about 5 children, which is a usual estimate. 



It is unnecessary to prolong these remarks by considering 

 the minor corrections to be supplied on account of the hypo- 

 theses not being strictly accordant with observation. The 

 two most important of these relate to populations that are 

 not stationary, and to the allowance to be made for in- 

 equality in number of the sexes. There are others hardly 

 worth even the trouble of describing, being utterly insensible 

 m rough work. 



The general results are that kinships fall into three 

 distinct groups : — (i) direct ancestry, (2) collaterals of all 

 kinds, (3) direct descendants, and that the number of in- 

 dividuals in each specific kinship in these classes is re- 

 spectively I, d — 3, and d. .\lso that d = 2i may be accepted 

 as a reasonable and not infrequent value. To determine the 

 number of individuals in each general kinship, the appro- 

 priate tabular number must be multiplied by the number 

 of species that the genus contains ; thus there are two 

 species of aunts, me si and fa si (mother's sisters and 

 father's sisters), each of which has the tabular number of 

 d — i; therefore the average number of aunts is twice that 

 amount, or 2d— i, which, in the above case of d = 5, is equal 



.to 4. 



Francis Galton. 



The Mendelian Quarter. 



A FEW weeks ago we heard in Section D at the Cam- 

 bridge meeting of the British .-Association a paper by Mr. 

 A. D. Darbishire on the bearing of his experiments in 

 crossing Japanese waltzing and albino mice on Mendelian 

 theory. He told us that on that theory we should e.xpect 

 a quarter of the offspring of the hybrids to be albinos— and 

 we found them albinos— and a quarter of the offspring of the 

 hybrids to waltz— and thev did waltz. Somebody protested 

 sotto voce, and Mr. Darbishire added "a rough quarter." 

 Since that meeting I have been looking up the rnatter, for 

 the point seems to me of great interest, and this is what 

 I find in a recent paper bv Mr. Darbishire in the Manchester 

 Memoirs, " On the Bearing of Mendelian Principles of 

 Hereditv on Current Theories of the Origin of Species," 

 vol. xlvi'ii. p. 13 : — " Let us consider the offspring of hybrids 

 .... Secondly with regard to their progression, we should 

 expect to find' 25 per cent, waltzing mice : this is very 

 roughlv what happens ; . . . Now let us look at the off- 

 spring' of hybrids from both points of view at the same 

 time: one mouse in every four is an albino; one in every 

 four is a waltzer, so we should expect one in every sixteen 

 to be an albino waltzer. Now these albino waltzers are 

 new things . . .," and then Mr. Darbishire tells us that 

 he has been unable to get offspring from them. 



Here, from a quarter, we have got to a quarter " very 

 roughlv," but still " one mouse in every four is a waltzer." 

 I must confess that Mr. Darbishire 's " rough quarter " 

 excited me to look further, and these are the words I find 

 describing some actual experiments on these mice : — 

 " Wallzim; occurs in only 97 out of the 555 individuals 

 resulting from the union 'of hybrids. When we compare 

 this with the number of pink eyed individuals (131-134) or 

 of albinos (137) we see that the proportion of waltzing 

 individuals cannot be regarded as a possible quarter. The 

 probable error of the expectation that a quarter of the 

 individuals will waltz is, on the Mendelian hypothesis, 

 06745 v/5XjX555 = 6-88 only, and the observed deviation 

 is i3875-q7 = 4i-75, the odds against so great a deviation 

 being rather more than 50,000 to i. As the result here 

 obtained differs from Mendelian expectation in the same 

 direction as that already obtained by von Guaita and to an 

 extent consistent with the agreement of both, the evidence 

 that the waltzing character does not segregate in Mendelian 

 proportions is very strong.'^ 



The sentences in italics are not in italics in the original. 



NO. 1822, VOL. 70] 



