78 Cooperative luvest igatiom on Plants 



fraterual capsules aud not fraternal plauts, — at least iu the case of Stigmata as 

 character. 



Let r = fraternal correlatioii uf plant ineans and R = correlation of capsules on 

 brother plants, p = homotypic correlation. Let i = a summatiou within a fraternity 

 and 8 tliroughout a population. 



Let z = number of Stigmata on a capsule, and m the mean number of Stigmata 

 on a plant of n capsules ; let a bar over a letter denote the mean value of that 

 quautity for the whole population and subscripts 1 and 2 refer to a pair of brother 

 plants. Then 



z = m + X 



S {z) = nm 

 S{z)=S{nm)\ 



where vi is the weighted mean of the plant means. 



■ and z = III, 



z- = iit- + 2nix + X-, 

 2 (z-) = nm- + 2 (x') = nm' + nai-, 

 if cTx^ = Standard deviation of an array of capsules on the sarae plant. 

 Hence o-^ being the Standard deviation of all capsules 



"' a{n) " Ä(n) "" ^ Sin) ' 



The first two terms on the right-liand side form a-,„-, the Standard deviation 

 squared of the means, the third terni is the mean square of the Staudard deviatious 

 of the arrays of capsules on the same plant = o-/(l —p'-'), if p be the homotypic 

 correlation. Hence we have 



p-'o-j- = o-,„-. 



Next 



Rtri- = c 



S{z,z,) 



S{>hn„) " 



(S(ni?ij) 

 _ )S(n,»!j??!i7»o) _, 



/Si(ni»!.,) 



Thereforc 



r = Rjp-. 

 Here of course 



^f(n.,z,) ^ <b'(»,ga) 

 Ä(n,»g S{n,iu)' 



_ _ S{niiumi) _ ;S\/(,;«.^;h.J 

 "*- S{n,n,) - S{n,n,) ' 



