MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 283 



TIT * (" ( l + 1) \ 



Mean x = - ....... (Ixxvi.), 



mi + m* + 2 



mr i '"i^ \ 



Mode or, = ......... (Ixxvii.). 



w, + m, 



. J> (in, w.) . ... . 



rt = a: a:, = . - ...... (Ixxvin. ), 



(/, + ) (?n, + w, + 2) 

 J> (/, + 1)(wi s + D 



-- - 



' . . . . (lxxx>) . 



/(, + ])(m, + 1)J 



For the moments we have 



//'(//(, + l)(w., + 1) 



^= - Hffr + i) ...... r ...... (IXXXL) ' 



L'6 :; (m, + 1) (w s + !)('".. - 'i) /, 



Ma= " r : i( , + T K 7T^r 



36' (w i + 1) (w, ^ 1_)JOiJ- J ) (/"..^1 )Q- - <J) +_-} /i \ 



^= H(r + l)(r+~2)(r+3) 



Lastly, for the mean and modal frequencies of ^ Sx and T/, 8.T, we have 



S (,, + ,,) -S (,,)-S (,,)! . . (1.X.XXV.). 



where 



Let 



J6 / ,. \nii / () . \MZ 



,(f) ^-T) ^ 



where 



^ r(<i + M., + 2) 

 Tr7ir+ i)r(n, + i)' 



then I (m u m. 2 ) = ?/, and we easily find, by the fundamental property of T functions, 



that 



2 o 2 



