138 THE QUANTITATIVE METHOD IN BIOLOGY 



Complete expression of the events under consideration (seeds) 

 may be obtained in the following way : — 



Each pair of primordia (factors) is taken separately. With 

 regard to the first pair, the four possible events are obtained by 

 expanding {D + Ry. (See § 104.) With regard to the second 

 pair, the four possible events are obtained by expanding {d + r)^. 

 Since the four compound events of both groups are combined 

 two by two, the sixteen possible combinations are obtained by 

 multiplying (D + R)^ with (d + r)^ (see §§ 94, 95) — viz. 



(D + R)^x(d + rY = 

 D^d^-^2DRd^ + 4DdRr + 

 D^r^+2D^dr + 

 d^R^-h2DRr^ + 

 R^r^ + 2drR^ 



In this way, the difference between J and $ being neglected, 

 the 16 sorts of seeds are reduced to 9 sorts. (Compare 

 §95.) 



Remark : The 16 possible events (sorts of seeds) are comparable to the 16 

 events which may take place when two dice with four faces, for instance, two 

 regular tetrahedia. (triangular pyramids) I. and II., are cast successively, I. being 

 S and II. being 9 , the faces of I. being marked Dd, Dr, Rd, Rr, the faces of II. 

 bearing the same marks. According to § 10 1, the possible events are cal- 

 culated a priori by expanding {Dd-\-Dr-\-Rd+Rr)^} If both tetrahedra are 

 cast simultaneously, the distinction between I. and II. (sex) disappears and the 

 possible events are 9 in number. The result is the same as with the first 

 method. 2 



From a biological standpoint the example of the tetrahedra contains perhaps 

 (? ?) something more than a method of calculation. In that example I suppose 

 that only two sorts of germ cells exist, one 6 sort and one ? sort, each sort 

 containing all the primordia of both parents and being thus completely 

 hybrid. According to the classic hypothesis, eight sorts of germ cells (four <J 

 sorts and four ? sorts) exist. On the other hand, we know (see § 38) that 

 a combination of properties may be segregated into primordia in several ways 

 without dissociation among the germ cells. In the above example of the 

 tetrahedra we may find perhaps a starting-point for a new explanation of the 

 Mendelian facts. . . ? 



One feels that there is in the Mendelian hypothesis of segregation, however 

 ingenious and useful it may be, something artificial which is rather incon- 

 sistent with the ways of nature. 



From the nine terms of the above polynomial (p. 138) we 

 obtain the same information as from the trinomial in the first 

 experiment. It is useless to repeat here the explanations given 

 on p. 120, (i)-(7). I limit myself to the following examples: — 



(Compare (3), p. 120.) From the number of terms we know 

 how many sorts of seeds are found in the Fg generation : nine 



^ Remark : The faces Dd, Dr, Rd, Rr represent compound events, the simple 

 events D, R, d, r corresponding to causes or factors (§ 97). The compound 

 events are combined in their turn two by two when the tetrahedra are caist. 



^ This may be easily demonstrated in the following way : — 



{D+Rfx{d+rf=i(D+R) {d+r)f 

  i{D^R) id+r)f={Dd+Dr+Rd+Rr)^ 



