in the development qf{l-\-x)~". 115 



" One follows the ordinary track ; showing that, if the theo- 

 rem holds for {l+x)~"'^\ it will hold likewise for {l+x)-\ 

 And this is readily proved by means of the fundamental pro- 

 perty of the binomial coefficients; viz. that the algebraical 

 sum of the coefficients of a?'"^ and .r*" in the development of 

 (1 +x)P is equal to the coefficient of a^ in the development of 

 {l+x)P+K 



" My other method having something peculiar in it, I shall 

 give it in full. Using S to denote the sum of the first m terms 

 in the development of (1 — <r)~", and R the remainder after S, 

 we shall have 



^^ l-(l-.r)».S 



Now, when we come to examine the numerator in this value 

 of R, we find that it contains only powers of x, from a?™ up to 

 ^m+n-\^ R may therefore be put into the form a;'"./(l — a)~^ 

 /{x) being used to denote a series of n terms proceeding ac- 

 cording to positive integer powers of jr, from x up to a;". 



" We have now ascertained the form of the remainder about 

 which we are inquiring, and it will be easy to determine the 

 coefficients inf. For this purpose let us take the equation 



{l-x)-"=l + na;+ ^^^^x^-i- ,.. + AnX"'-'-hx'^.J{l-x)-\ 



and in it interchange x with 1— a?, and m with n. Then we 

 shall have 



x-m=z\-^m{l-x)+ ^^^+ ^) (1 -xy+... + A„{l-xy-' 



where/' ^~' stands for a series of powers of x~\ from a;~^ up 

 to .a?""*. Multiply the last equation by a?'" ( 1 —a?)"'*, and it will 

 become 



(1 -x)-" = a:'» -{"(1 ---^)-»+ »?( 1 -3?)-''+' + ^^^^(l-a?)-"+2 



+ .., + An{l-:t)-'^ +x^f'x-K 



On comparing the several terms of the two finite developments 

 thus given for (1— a^)~", it is obvious that we must have 



and 



Rssa;"' <{l -a?)-« + w(l -a;)-»+' + ^^^^^^ (1 -x) «+2 

 l_ 1 « J 



12 



