Mew Mgehraic Series. 281 



f a{a-^k){a+^k) - - -(«+(F-1)^)^ 

 + -y- 6(a+A-) {a-{-2k) . . (c+(p -2)^) 



«^ + ' . - - &c. > 



+^-^(a+kUb+k) - .(6+(p-.S)^) 

 ^ +6(6+A;).(6+2^) - - ib + {P-2)k) J 



And the Second. 



f a(a-f-A;) (a + 2A;; - - - (a + (;?-2)A;) "^ 



+^(6+^).a(«+A;) -- (a+(p-3)^) 

 + - - - &c. 



bJ P P—2 \. 



-^^a{b-^k){b+2k) - .(b + {p-2)k) 

 ^{b+k){b+^k) - - - (6+(p-l)A:) J 



Fn these expressions it is evident that the mullipHer of a, 

 in the first of these two parts, is that which would become 



zp — 1 

 theco-efScientof-j-^ 7:^7~:'\, when aischanged into a4-^; 



and that the multiplier of b in the second, would become 

 the same co-efficient, when b is changed into b-\-k<. The 



co-efficient ofr-r — ~~/~_-, \ being therefore, reducible to 



the form 



(a+b){a+b+k)(a+b + '2k) - - {a-\-b + {p-2)ky, 



the multipliers of a and b will each become 

 {a-\-b+k) (a+b+2k) {a+b + 3k) - - (a-^b+{p-))k) 



Both these multipliers being united, the co-efficient of 

 zp 



will therefore be {a+b) {a-\-b+k) {a+b+2k) 



1.2. .p 



ta-\-b + {p-\)k). 



Vol. VIT.— No. 2. 36 



