336 REPORT — 1872. 



also al(u^w^ uj^^ ^ = '^~±{a^—a^) 



where the upper or lower sign is to be talcen according as a is less or greater 

 than /3, and where S refers to v, and i'=l, 2, 3. . . .w. 



Now let K=J,^20?^^^^^;ZJV^P)' 



then 12 2v 



K,-K,+K^-....+K^=0, 



for .'=1, 2, 3 n. (See Jacobi, Crelle, 13.) 



Moreover, let 



2.-1 2c (^"20 ?('^)^^' 



•^ "2c- 1 

 _ 2c-2 2c-l r'^2c-l '?{x)clx 



•^ 2c— 2 



then also we find 



K=Ki + K^_ ,+ .... K^,,-»-K,i, 

 i=K.2+K,3+.... K,,-'-K.P 



4 



5^.=I^..,3+-- •• +K„„-^K,^-iK,^2. 



2a _ _ _ • 



V y, 1 v, 2 I", ft ' 



then the following four fundamental formulae hold good, where we make use 

 of the symbol jjla. to denote zero when a- is less than j3, and unity when a is 

 greater than ft ; — 



(1) 



(2) 

 «^(Wi • • • • )^ 



