74 Mr murphy, ON ELIMINATION BETWEEN AN INDEFINITE 



The systems of the first equations thus obtained may be written in 

 an inverse order thus, 



&c. = &;c 



whence «„, «„_i, a„_2, &c. are successively known. 



We have supposed all the functions to be of n dimensions, for 

 these necessarily comprise all of lower degrees. 



Example : 

 To expand unity in terms of the functions 



af, {x-^hy, (a; + 2A)", (a; + «A)°. 



Put l=«o*" + «i(^ + ^0° + '''^(^ + 2^)" + + «„(a;+wA)"; dividing by 



a;", we get 



= ao + ffi +«2 + + «„, 



1= ai?'i + Oag-'s + + «»<?'», 



where we have o'„ = A. {waf"*.»i-l — , ^ ' hm^af-^+ \. 



^ 1.2 



Divide now by g'l and we obtain 



= ai + 2a2 + + wa», 



1= «2g'"2+ + a„g^'„, 



where in general g'"„ = A^ |— ^ — ^— ^ a;""* {m^ — m) + &c. > . 



This process is easily continued, and we obtain successively the 



equations 



= 1.2a2 + 2.3a3 + (»— 1). w«„, 



0= 1.2. 3 03+ {n- 2) (71-1) nan. 



and lastly, ^= 1.2.3 «a„ 



