290 Proceedings of the Royal Irish Academy. 



and that 



^2«+i = r"^' + 2w .y-«-^ . ^ + ^ -^ — -i?^"-' . t 



(2^-2)(2>^-3)(2^-4) (« + 2)(^ + l)^ 



1.2.3^ *^ '^ ■ ■ ■ 1.2.3 '^ '^ 



+ (w+ 1) .^^" . . . . , (2) 



eaeli of wliich series consists of n ^■\ terms. 



Conversely, the sums of the series (1) and (2) are known; for 

 they are, respectively, lun and ^2«+ij i- e., 



and 



(a-'^+2 - yS2" + 2) ^ (a - ^). 



Finally, 7^,^ is expressed under another form, viz. : — 



T At — V) 



+ f • K-2r ; 



which may be expressed symbolically thus : — 



II. 



If x"' = pi.x'"-'^ + p2-x"'~- + pi.x'"-^ + . . . + Pm, to express 

 ic" {n > m) in form 



Ii.x'"-^ + Po.x'"-"" + Pz.x"'-^ + . . . + P,„. 



A particular case is first considered. If x'^ = px^ -f qo^ -^ rx \ s, 

 to express ^" in the form 



la? + Qx^ + Ex + S. 



"We may briefly indicate the solution thus : — 



Multiplying both sides of equation x^ = pa? + qa? + r;K + s by a;, and 

 arranging the terms, we shall find 



oc' = h^.a? + {qhi + r)a? -i- {rhi + s)x + sJh . 



