42 DR GUSTAVE PLARR ON THE 



If we consider the second of these factors, we see that R u contains 



■■(27i-2u).' 



m2n-2u) x 



U(n-s-2u) 

 Hence 



x n - 1 - 2«tt( 2 n - 2u) _ d n+ V" " 2 " 

 H(n — 8 — 2u) dx n+s ' 



and we may write 



(-1)" 1 (-l) u n v i-i 



2 n llull(n-u) 2 n Un 1"/ +1 



Hence with the signification of r, r f , 



Now the sum in the second member, considered in itself, and in which the upper 

 limit of u may be extended to u = n, represents 



(x 2 -l) n , 



but the differentiation in respect to x causes the terms to disappear, in which u out- 

 passes r or /. We have then 



We put, as it is usual, 



= d n (x 2 -l) n 

 n ~ 2*Ilndx n 



and then 



[r ''u]R(uy— - =<g^ 



The same expression holds for 



where x is replaced by x / . We have then 



whereas Z„, expressed in 2 as at the beginning of § I., is 



7 _ <w- l) w 



"~ 2 u Ilndz n 

 1 10th being the known expressions for Z m . 



