MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 301 



In general none of these correlations vanish, and their values must all be found 

 before the errors and correlations of the chief characteristics of the frequency can be 

 found. 



The following results, easily obtained by aid of the relation tan <f> = v/r, will be of 

 service 



$1 = C ^{S;+tair^-2tan^SA,) .... (clii.), 



B^-^^S, -tan^R,,} ......... (cliii.), 



' *< 



B*, = ^{2,11,,- tan f,} ......... (cliv.), 



E,,,,, = -^{SA,, -tan^Ee,} ........ (civ.), 



T2.$ 



R, i = ^{SA,-tan^,.R,,} ........ (clvi.). 



i ,f 



By (cxxvii.) and (cxxxi.) if x be the mean size of organ, 



A,x = A/i tan d>Aa - - Ar/>. 



cos- <j> 



Ao- A'< , . A/- 



= - + tan (f>k(p jf - 7 



(7 ft i' I 



Hence 



^ = ^ + tan^ ffi + c ~Z,-2 tan <^2 K R,., - ^ S fc 2^ t 



X^R a , . (clvii.), 



= + tan^ ^ + S; + tan 



T^S.R*, .......... ". . . . (clviii.), 



y -~~ -L 



and 



ET \ 2 a S A/, tan v^ ^a^Rn^ I f nl , J.V V "R - tan' 2 AS? ~$ T? 



- = =r^r < - --- - X ;dhr T tan ( P-* 2 '* it ** t ' an < PA l ^f.^a 



2t a z,^. L fl cob 9* 



tan < , ft V ^ Ti Mix ^ 



^^ ' 



From (cxxx.) we have, if S* = skewness, 



