118 Mr. Lubbock on some Elemenlary 



= A {x—x^) [x—x<^{x—x^ ('^— <37^)> 



ou A ne depend pas de x^ je dis qu'on aura 



^ Jit , C(ao+ ai«+»»»+^n«") a /( ^i«)+(co+Ci«4->.-+c>» «"") V(^g«) ? 



oii C est une quantite constante et r le coefficient de — dans 

 le developpement de la fonction 



fx , ^ (ao+aiX + ...+anx")A/(^ia:)+(Ci+Cig+.»+^ma?"*)V(^2x) ^ 



suivant les puissances descendantes de x, Les quantites Sj, Sg, 



6^ sont egales a+l ou a— 1, et leurs valeurs dependent 



de celles des quantites x^, x<^ x ^" 



Suppose generally, 



fijT = aQ-\-a^x + ac^x^ +fl„j:" 



^^X = Cq +C^X+CciX'^ +c^x*". 



The values of sj, gg, 63 e/a, are not arbitrary, they de- 

 pend upon the magnitude of x^, x^ j;^, and this is deter- 

 mined by the equation 



$x ^/ <p^x =3 sQ^x \/ <p^j?. 



Moreover, iffx is divisible by x—u, fa. = ; and if (fx)^ is 

 of less dimensions than <px, r vanishes, and 



si^a^^ + Sci^x^ + Sf^^x^ = C, 



In the following examples A is made equal to unity, which 

 is allowable. 



It is intended here first to apply the theorem to the integral 

 In this case (p^x = l+x-hx\ ^^x =z l—x 



1 -^-X + X^ — C^ (1 —X) = {X — X{) (X-X^) $X =z I $^X Si c* 



Equating the coefficients of powers of jr, 



