246 ON THE INSUFFICIENCY OF TAYLOR's THEOREMy 



whence 



<^x — (ax + a') = 

 or 



^x = ax + (a' + c) = ax + a'. 

 And, in like manner, if ^ x is a function that has its third differential coeffi- 

 cient equal to zero, we may write 



d^ ^<?)X — (ax'' + a^x + a'')] = o 



d i^x — ( ax'' + a^ n + a'^) I = c 



X ^ ^ 



^x — (a x^ + a'x + a'') = ex + c' 



^x = ax'' + a^x + a^^ 

 whence it appears that ax^ + a' x + a^^ is the only function, excepting those of 

 the class already mentioned, which has its third differential coefficient equal 

 to zero. And, proceeding in like manner, we find 



S a ' • • ■^ x"^ 



z = 0|ij — 1 



to be the only function which has its n*"* differential equal to zero, the exception 

 above mentioned being still kept in mind, and which is removed by substi- 

 tuting for each of the constants, functions that are constants between the limits 

 assumed, and of arbitrary values beyond those limits. 

 Now suppose 



d^Fx = d"^x 



we can divide ^ x into two parts, F x and 4' x, and write 



d"Fx = d"(Fx + 4'x); 



X 



but this equation gives 

 whence we conclude that 



4)X = Fx+ S a"'''x^ 



z = 0|n — 1 



is the only function that can give the equation 



dn4)x = d^F(x) 



X 



or the terms to the left of d" F x in Taylor's Theorem become prime on add- 

 ing the customary function, and substituting for the constants it contains, func- 

 tions of the class we have described." 



The occasion of Taylor's Theorem failing for special values of x is mani- 

 festly by such values destroying M in a term 



