8 Mr Olaisher, On a formula of Oa^tchy's for the [Nov. 6, 



since/' is an uneven function. Thus -^=— -r- and therefore 

 = 0, so that u is independent of a, and we have 



/>(--s)t=//(-"'^ ^''' 



of which (7) is the case when a = 1. 

 It then follows that 



/;/(.)..=/;/(.-i)<^.=/;/(.-i)^f 



which is (5), and the proof proceeds as before. 



§ 4. But having proved (7), the more interesting procedure is 

 to divide the equation (6) hj x -{- x~^ previous to integration^ viz. 

 we have 



\ x/ 3! \ xj 



and on multiplication by <f>(x — x~'^) and integration the right- 

 hand side becomes nP^ + &c., i.e. the right-hand side of (2), while 



j (x'''-'-x"'-'+x"'-'...+x-'"^'-x-^'''-')<p(x-^]dx 

 all the other terms destroying one another, since by (4) 



&c. = &c. 



Thus, regarding (7) as the fundamental theorem, we see that 

 we obtain a formula for 



/>'>h3^-' 



