274 Preponderance of Males and Females in Fish 



APPENDIX. 



(I) Calculation of the change in constitution and sex-ratio resulting 

 from the continuance through a number of generations of a modifying 

 factor transforming a proportion of zygotes of male chromosome-consti- 

 tution into somatic females. 



(A) When ^ is XY, % is XX. 



Suppose tRat ^ of the XY zygotes in each generation are trans- 

 formed into females. 



If the constitution of the nth abnormal generation be UnXY -. IX X, 

 then pUn X Y will be transformed into females, and the sex-ratio will be 



(l-p)w„(ZF)V:l(XX)-Hi>i/„(ZF) ? = (1 -jtj) i^ncf : 1 +i>^« ?• 

 Of the gametes, the spermatozoa will be 1 X : 1 F. 

 The ova will be (2 + pUn) X :pUnY. 

 The constitution of the next generation will therefore be 



pUnYY (non- viable) : (2 + 2pUn) Z F : (2 4- pit„) XX. 

 Thus u.,.= %^,or,i{q='-,=^l-±^ = l+ "' 



2 + pUn ' ' ^ p' 2q + Un Un + 2q' 



Now Wi = 1, . ". ^2 = 1 + -, . o ' • '• Wa > Wi. 



1 + 2q 



Similarly u^ > u.2. However, u tends to a limit given by 



.■.uj + 2{q-l)u^=2q, 

 .'. u^ = l + s/q'^ 1 - q. 

 And the final sex-ratio is 



(l-p)^oc-</ '^ (g-l)^^ _ (g-l)(l + Vg+I-g ) 



\-\-'pu^.% q + ^^^ i + V^'-hi 



When the disturbing factor ceases to operate, the swung-back sex- 

 ratio will be 1 $ : Un ^ . The preponderance of males after a swing-back 



