MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 271 

 or 



Sfc -*j+i v ';7{(jTT)S) ...... < lvii -)-* 



Similarly 



S(ASA) 

 J*w = v v = 0, since R hj , = 0, 



~St^ 



or 



RAS,, = ........ . (Iviii.). 



We easily obtain, by multiplying (liv.) by A/>, the result 



We can now obtain R, ffS ,, for every A St is negatively proportional to the corre- 

 sponding Ap. Hence 



= {1 + SQrTiys"} 1 ........ ( lx> )' 



We next pass to the mean and modal frequencies as given by (xli.) and (xlvi.). 

 We have, by taking logarithmic differentials, 



where 



* If S' = -A^ - B3 + - B5 - - &c., then it is easy to show that i + / S' = (p + 1)' S. 

 Remembering that, if Si = a, q - , we easily deduce 



J = + ng r (jp + i)} - log (p + i). 



' + f j + (B 5 + 2 



+ 3(Sk.)'+j 



as far the 7 th power of the skewness inclusive. 



Very generally the probable error of the skewness may be taken as equal to 



\'2n! v/{l +3(Sk.) 2 }' 

 and it is always less than ^(3/2%), its value in tlte case of a normal frequency. 



