IX] and Spurious Allelomorphism 159 



depend on one allelomorph, not on two. To prove the 

 existence of complete coupling it would be necessary to 

 show that features elsewhere known to depend on separate 

 allelomorphs, could on occasion be linked in a complete 

 union. Whether such a state of things is possible we do 

 not know. There is no a priori reason for supposing that 

 it is impossible. 



The arithmetical series in which the numbers occur is 

 the only guide as to the nature of the process, and obviously 

 this is quite insufficient. The existence of the 7 : i systems 

 and of the 15:1 systems naturally suggests the possibility 

 that a system based on 3 : i may exist. We might then 

 arrange the systems in a series thus"^ : 



Gametes 



No coupling lAB lAb 



3 : I 2>-^B lAb 



7 : I TAB lAb 



15 : I iS^B lAb 



Hitherto, though some dubious indications of such 

 a series have been seen, there is no clear case of coupling 

 on the system 3:1. 



It is not easy to conceive any probable system of 



symmetrical cell-divisions or dichotomies which would 



produce the series 7 : i and 15:1. If the segregation 



of characters were not all completed at one cell-division 



we might of course imagine a scheme which would give 



the system 8 -f- 1 -H i + 8. thus • 



AaBb 



/ \ 

 ABb aBb 



/\ y\ 



AB Ab aB ab 

 after which, if the cells AB and ab each divided again 



* The F2 numbers resulting from these couplings are as follows : 



AB,Ab.aB.ab. 



3 •• I : I •• 3 41 : 7 i 7 : 9 



7:1:1:7 177 : 15 : 15 : 49 



15 : I : I : 15 737 : 3^ • 3^ = 225. 



If n be half the number of gametes needed to express the whole series of 



couplings in a given case, then the four F^ numbers are given by the 



formula 



3«^ - (2« - i)\2n-\ : 2« - I \r^- (2n — i). 



