.CAUCHY'S PROOF OF TAYLOR'S THEOREM. 81 



amounts to the assertion that there must exist some point It 

 between P and Q, such that 



are&PMNQ _L 



rL ' 



106. Suppose now that F(x) and f(x) and all their dif- 

 ferential coefficients up to the (n + l) th inclusive, are con- 

 tinuous between the values a and a + h of the variable x ; 

 moreover suppose that one of the two F'(x) and f (x) does 

 not vanish between the same values, also one of the two 

 F" (x) and /" (a?), and so on up to F n+1 (x) and f n+1 (x). 

 Then, by Art. 99, 



F(a+h}- F(a) _F'(a + Of) 



~ 



F'(a, + Of)-F'(a) _ F" (a 



F"(a + Of)-F"(a) _ F'" (a,+ Of) 

 f" (a + Of) -f" (a) f" (a + Of) ' 



F" (a + eji) - F n (a) _ F"* 1 (a + OK) 



where O v 2 , ...... n , 6, are all proper fractions. 



Let us now suppose that F' (x), F" (x), ... F n (as), f (x), 

 f" (x), . . . f n (x) all vanish when x a ; then from the above 

 equations 



) - F(a) ^F n+l (a + 0h) 



-/(a) -/"(<* + Oh) ' 



107. If we take f(x) = (x d) n+1 we find that the requi- 

 site conditions are all satisfied ; that is, f(x) and its diffe- 

 rential coefficients are continuous, and the differential coeffi- 

 cients do not vanish between the values a and a + h of the 

 variable; also all the differential coefficients up to the 71 th 

 inclusive vanish when x = a. And 



f n ^(x)=\nl, /(a)=0, f(a + h) = V + \ 



Suppose then that F(x) and all its differential coefficients 

 are continuous between the values a and a + h of the varia-? 



T. D. C. G 



