72 TAYLOR'S THEOREM. 



92. Suppose / (a + x} and its differential coefficients up to 

 the (n+ I) 01 to be continuous between the values and h of 

 the variable x. The expression 



vanishes when x = h if R = 

 \n + 



Suppose R to have this value which we observe is inde- 

 pendent of x. 



The expression (1) also vanishes when x = 0. 



Hence, by Art. 91 the differential coefficient of (1) with 

 respect to X must vanish for some value of x between and h ; 

 suppose #j that value, then 



vanishes when x = x t . But (3) also vanishes when x = ; 

 hence there is some value of x between and x l for which 

 the differential coefficient of (3) vanishes. 



Continuing this process to n + 1 differentiations of (1) we 

 find that / n+l (a + x} R is zero for some value of x between 

 and h ; let this value of x be 6h, where 6 is some proper 

 fraction, therefore 



Substitute this value of R in (2) and we have 



We may now put a; for a in this equation, since there has 

 been uo restriction in the value of a, except that all the quan- 

 tities are to. be pirate, thus we obtain 





