78 CAUCHY'S PROOF OF TAYLOR'S THEOREM. 



F(a + 8g) - F(a + 2a) = F' (a + 2a) , 

 \o + 3a--a + 2a~"'a+2a + ' 



-/(a+A-a) /'(a + A-a) 

 where 7, 8, . . . /*, all diminish without limit when a does so. 



Since the fraction in (2) always lies between the greatest 

 and least of the series 



. fi . , 



' % 



A-a) 



it must lie between the greatest and least limits towards 

 which these tend ; that is, it must lie between the greatest and 



F'(x) 

 least values which ... , . can assume between a and a + h. 



f fc) 

 F'(x) 

 But as ,.,,{ , in passing from its greatest to its least value 



/ 00 

 passes through all intermediate values, there must be some 



proper fraction 0, such that 



-F(a) F'(a + 6h) 



101. Suppose f(x)=x a] therefore /' (ar) = 1. 



The conditions required to be satisfied by f(x) in the 

 enunciation of Art. 98 are satisfied. And as f (a + h) = h, 

 and /(a) =0, 



we have F(a + h)- F(a) = JiF' (a + 0A). 



This simple case of Art. 98 might of course be proved iu 

 the same manner as the general proposition was established. 



102. The result of Art. 101 may be applied to shew 

 that an expression independent of x is the only one of which 

 the differential coefficient with respect to x is always zero. 

 For suppose F(x] a function, such th&t.F'(x) is .always zero; 



