280 Kew Algebraic Series 



zp 

 efficient, or 



l.'2.,.p ' 



Now for the first of these co-efficients we have 

 «(a+^)(a-f2^)(a-f3A;) - - - («+(?— 2)^) 



»— 1 

 -h-^Y~^- «(«+>^)(a4-2A;) - - {a+(p—3)k) 



+ . - - &c. 



p-l 



-{■■—- a. b(b-\-k){b-{-2k) . - (b-{-{p-3)k) 



-hb{b+k){b+'2k) - - - - . (64.(p-2)A:) 

 And for the second 



a{a+k){a + 2k) (a_|_(p_lj^) 



■^-b. a{a+k)(a-h^k) - - - - (a4-(p_2)A;) 



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

 + &c. 



+f ^y-«(«+^). 6(6+^) - - (6+(p-3)A) 



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

 +b{b-hkXb+2k) - - - . - (b+(p~l)k) 



P P — I 

 But as ^=-^-^+1 



p p — 1 p— 1 p — 2 p — 1 



P P~l p — 2 p— 1 p — 2 p — 3 p — 1 p— 2 

 T 2 3~~~ i 2 3~"^ r~* 2~ 



fee. 

 it is plain that the above co-efficient can be decomposed in- 

 to two parts, the first of which is 



