50 PROCEEDINGS OP THE AMERICAN ACADEMY 



Let now 



/i(0) = [(<? - ft)- - (ft - ft) m ]» [(fl - ft)- - (ft - sO m p . . . . , 



the remaining factors corresponding to the remaining latent roots; 

 then for g = ft , 



fx(9) = /ifa) =t 0, f^ff) = 0, f{\g) =0, .... f^~\g) = 0; 



for g = ft, f x (g) = 0, and likewise its first n — 1 differential 

 coefficients ; similarly, for g = ft, f x (g) and its first p — 1 differ- 

 ential coefficients are all zero, etc. Therefore 



/i(0) _ v m r 1 

 /i(^) _ ^ lrLr ' rJ ' 



It may be shown in like manner that 



A (9x) 



(6 - ft) = &- r "» [r, r + I], 



£|I(0_ft )2 = Sl V[r,r + 2], 



^(^-ftr-^ci, H. 



Similarly, if 



then it may be shown in like manner that 

 4|®. = S£ [» + r, m + r], 



4^r(tf-^ = &"; 1 [i»+r,»+r+l] l 



/ 2 Gfe) 



4t^ C^-^)"- 1 - [» + 1, m + ft], 



72 (.#2,) 



etc., etc. 



