o/ Taylor's Theorem. 137 



then the theorem in question affirms that 



/,=/+ n .fh + (n . )y^ + (ft . ffY^ + ■ • • • 



On making a = l, b = .Vy c = 0, d = . . . , the theorem becomes 

 Taylor's. To prove it in its general form, let 



71—1 



<^a; = ax^ + nbx'^~' ^ + n —^ — cx^~^ + . . . ; 



then, on substituting x + h for x, (f>x becomes 



w — 1 

 = aix"' +nbiX^~'^-{-n-—^ — CiX^~'^+ .... 



Let A become h + SA, then obviously 



But we may obtain the new values of «i, 5i, Ci, . . . cori'espond- 

 ing to the change of h into li + 8A, by substituting in <^x first 

 ar + 5A and then x-\-}i for .2?. 



The effect of the first substitution is to change a,h, c,, ,, 

 into a + ha, h-\-hh, c + Sc, . . . , where 



ha = 0, hh = ahhj 8c = 2hShj Bd = 3cBhj », .. 



Hence the increment 



consequently 



Hence, if we write 



/i=/+BA + CA2 + D7i3 + ..., 



we shall have 



B + 2CA + 3DA2+ ... 



=nf+aBh+2nch^+ .... 



Hence 



B=a.f, G=i{n. yf,-D=^{a.)f...; 



* Or without introducing (fiX, tlie equations between «j, 6^, c^ . . . and 

 a,b,c,... show by direct inspection that the effect upon the former is the 

 same, whether we augment h by dh or h,c,d.,. respectively and simuha- 



d 

 neously by ahh, 2bdh, 3cdh, ... so that ^ /i = ^fu as in the text. 



