NUMBER OF, UNKNOWN QUANTITIES, 73 



SECTION III. 



Amplication of the Second Principle. 



To expand a given function of x as P, in terms of other given 

 functions 



Qo, Q., Q. Qn, 



all being supposed of n dimensions in x. 



Let P=aoQo + «iQi + «2Q2+ +«nQ», 



where a^, ffi, Oa «» are constants to be determined. 



Divide all the functions by Qo, and let the corresponding quotients 

 be respectively 



P', Qo, Q'l, Q.....Qn, 

 and the remainders 



p', g^o, q\, q'i q\- 



Then by attending to the second principle, we have 



P' = «oQ'o + «lQ'l + (kQ2+ +«„Q'n, 



p' =aoq'o + aig-'i + «25''2 + +a„q'n, 



when we obviously have Q'o=l and §''0=0. 



Dividing the last equation by q'l and using a similar notation, we 

 get in like manner 



P'=«.Q". + «2Q"2 + ««Q"„, 



p"= aiq"i + (hq"2 + anq\, 



where Q"i = l and q'\ = 0. 



Divide the equation last obtained by q"i, and we obtain 



P"' = a,Q"',+ +a„Q\, 



p"'==a»q"', + +«„^"„, 



in the latter of which equations the first term = and in the former 

 it equals unity. 



Vol. V. Pakt I. K 



