344 REPORT — 1873. 



It will be sufficient if we prove the first and last of these formulae. The 

 first is proved by taking the values of K^,^ ^, J^ ^ already given in sections 

 1 and 4. 



2c— 1 2c 2c' — 1 2c' 2c'— 1 2o' 2c— 1 2c 



2(K,A,,-K,,,J,,,) = S{(K-K)(J^-JJ-(K-K,)(J^-JJ} 



2c— 12o' — I2c' — 12c— 1 2e 2c' 2c' 2c 2c 2c'— \ 2c'— X 2c 2c— I 2c' 2c'— I 2c 



= S(K J -KJ) + i;(KJ -KJ)_s(KJ -KJ)-2(KJ -KJ ^=0 



2c-\ 2c \ 2 3 2c-2 2c-l 



S(K J' -J K' ,)=-i^ {(K -K)(.T -J +J -J 4- +J -J -) 



2c— I 2o 1 2 3 2c-2 2c-l 



-(J.-JJ(K -K +K -K + . . . .K -KJ} 



TT 



Section 7. — Let now 



w' = »-,Iv' ,+ 7- K' ,+ + r K' , 



e =w, J i + ??i J„ 9+ . . . . +m J , 



<='-:j;.i+'-.j;.2+ •••• + »•„ j;,„; 



then we shall have 



Al(M, + 2,u, ....) = e~^^''''''^"''^''''Wf,... ), (1) 



Al(M+2w\t....) = e~^^''''^^"''^'*'''''^'.(Al(M^....). . . 



We shall prove the first of these formulae. 



We easily deduce from equation (1), section 5, that 



Al(z^, + 2mK,. . . . ) = e-2^''"'^>''+"^^^)Al. K. . . .), 

 where m is an integer. Hence 



A10t, + 2mK, + 2rK,....) = 



(2) 



