308 Combination of Linkage Values 



It is believed that the above method of estimating distances will 

 prove of considerable value when applied to comparatively long chromo- 

 somes in which factors are sparsely located, such as the second and third 

 in Drosophila, since there is no reason to suppose that the relation 

 arrived at between distance and cross-over value is peculiar to the sex 

 chromosome in Drosophila. The results of investigations on these chro- 

 mosomes should go far to confirm or refute the theory. 



Outside Drosophila the best series of results on which to test it are 

 those of Altenburg(9) with the three factors M, S, and G in Primula 

 sinensis, quoted by Punnett(lO) in a recent paper. Here the cross-over 

 value for M and 8 is 11-6 %, for M and G 34-0 %, for S and G 40'6 7o, 

 each result being based on 3684 individuals. By Table II the distance 

 SM is 12'1 centimorgans, MG 40*9, and hence SG is 55*0 centimorgans 

 (assuming the loci to lie in the order SMG). Hence the cross-over value 

 for S and G should be 404 °l^, the observed value being 40*6 °/^, a very 

 nearly perfect fit. The addition formula gives 45*6 °/q, Trow's formula 

 37"7 7o- The probable error of the calculated result is '64 705 of the 

 observed '55 7o- Hence the probable value of their difference is *84 7o> 

 and though the close agreement is accidental, both the alternative for- 

 mulae are impossible. 



In the case of Punnett's(lO) results for sweet peas the agreement is 

 also good, but owing to the closeness of the linkage, the three formulae 

 give nearly equal values. There is, however, no reason to suppose that 

 Table II does not represent with fair accuracy the relation between dis- 

 tance and cross-over value in all organisms, though the absolute value 

 in fifi of the unit of distance, or morgan, is presumably different in 

 different cases. 



Summary. 



By a consideration of the observed gametic ratios of the sex-linked 

 factors in Drosophila, the following results, among others, are arrived 

 at: 



1. li A, B, and C are three factors lying in a chromosome in that 

 order, and if m is the cross-over value for A and B, n that for B and C, 

 then the value for A and G lies between m + n and m + n — 2mn, being 

 nearer to the former when m + n is small, to the latter when it is large. 



2. A relation is arrived at, on the hypothesis that the chromosomes 

 are partially rigid, between cross-over value and distance, which permits 

 of the calculation of one of the cross-over values for three factors from 

 the other two, with a probable error of less than 2 °/^. 



