42 



The differential theorem may be satisfactorily established 

 by conducting the investigation thus : 



Let 



a — bx -\- cx^ — dx^ + &c. = s 

 ••. — bx + cx^ — dx^ + he. zz s—a (10) 



... _ 6 4.ca; + </a;2 + ea;3— &c. = ^^. (11) 



X 



Consequently, by adding these two equations together, and 

 representing the numerical differences b — c, c — d, d—e, &c. 

 by A, A', A", &c 5 there will result the equation 



- b-A.x-\- A'.x''-^".x^ + &c. = ^i^ (s-a) (12) 



.-. —bx-A.x'^ + A'.a;3- A'^.a;" + &c. = (a; + I) (s— «) = s' 



s' 

 .-. s = + a; 



x+l ^ 



that is, 



s = a - -^ + -^ [0-A.a; + A'.x-'-A^'.x^ + &c.] (13) 

 x-\-l x-{-l ^ 



And by treating the series within the brackets as the original 

 was treated, and so on, we shall finally obtain the transfor- 

 mation 



_ bx A.a;^ A'^.x^ 



^ - ^~ ^+1 ~ (s^+iy " (^+17 ~ 



or putting a = 0, and dividing by — x, we have 

 b — ex ■{• dx^ — ex^ + &c. = 



^ + 1 + {x+\f + {x^lf + (a;+l)^ "^ 



which is the usual form of the theorem. 



Now the preceding reasoning is inadmissible except the 

 proposed series be convergent ; that is, except rx^ approaches 

 to zero as n approaches to infinity, rx^ standing generally for 

 the w"* term of (10). For in (12), which results from the sum of 

 (10) and (11), this w'^ or final term, is regarded as zero, and 

 is neglected ; inasmuch as it is by this term that the series (10) 



