MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 287 

 In like manner from (Ixxviii.) 



M A6.fl 1 1 



- = + Aw, < - - --- 



d 1'iHi ni* w, -f- m., ?/?, + in* + 2J 



* 1 



+ m* 4- -J 



4- PI Am, 4- C'. 



say, where the numerical values of c, and c, can easily be found in any actual case. 

 Hence 



Again, from (Ixxix.) 



Ar A6 /I 



7 : : T 



Or AO-/O- = A&//> + f , Aw, + 



- = f- + eiS;,, + ^S:,, + 2^ 2 S,,,^,,, 2 -R, MiM2 + f ' S^,,,^,,,,,, 4- ^ S^K,,. (cxi.). 



Further from (Ixxx.) 



AS,__ J __!__ _l_ 



' " '- ' J 



4 



= /, Am, 4-/, Am,, say. 

 Hence 



;/aS Wl S )I1 ,E tolW ,} .... (cxii.). 



From the results given above we can deduce the effects on size, range, variability, 

 or skewness of a selection at random of any one of these four. 



Writing (cvii.) 



AM e = A/i 4- y l A& 4- g.. Am, g* Am, 

 we find 



R<r \ ^b^t^-bh f/i <2 i '<! v V T? 8* V ^ P _l_ /- S 1 V T{ 



MH7 = g ^r ^ 4- "5" * + y A 6 i Wl t tB!l --- ^ *s2,J*ta,, -| t^Zfci,,,, n,;,, (tl 



4- e^.^^Ri-, + e^,S;, - e^^^^, + ^S^B*, 4- ^/iS 4 S, % R AfflJ 

 4- ^S^SA^ - ft^S 2 ., ......... (cxiii.). 



