342 REPORT— 1872. 



Hence we may manifestly assume, if J be some constant, 



V 



2a- 1 



put in this Mj — Kj for u, and we have 



d log al(UAl ....)?„ 1 -r . ^^i 2a-l 2.-- 1 



I* 

 Hence, by addition, 



2a— 1 21/—! 2a— 1 iv-1 



2v-l 



and this must be true for all values of u: put for ti^, K^, and 



2a- 1 2o-l 



2J=a\(K,....).-aX(-K.. ...)., 

 or 



2a- 1 2a- 1 



J=aX(K,....X=J„ 

 which determine the arbitrary constant, and we have 



Section 5. — It may be proved by the help of equation A of last section 

 that the expression 



2^-1 2f-l 



is a perfect differential. 



Now let us define two new transcendents as follows : — 



d log, Al(ir.„ «,....)=- 2,R+ a\(i{, - K, .... )}du^, 

 and 



Via /a Ai/ \ 



Al(Mjlf,....) 



Combining these equations together, and making use of equation (1) of last 

 section, 



21/- 1 a 2i'— 1 a 



dlog^Al(u^n,. . . .)^=_s/,J,-J,+ aXK-K, + K,. . . .\}du^. 



a a 



Now putting «i+Kj, m^+K^,. . . . for tt^M^. . . ., we have 



<Zlog^Al(«^ + K^,w,+K^,....) 



= - 2,Tj,4- aX(«, + K, - K^ . . . . \}du^, 

 or 



fZ log, Al(i«j + K, ....)-d log, AH», . . . . ) 



