MISCELLANEOUS PROPOSITIONS. 405 



Multiply by ^(a), and put for <f>(a) and ^(a) their values ; 

 then 



F(a+ h) -F(a) -hF'(a) - ... - ^ F " (a) = E, 



[2 

 where R = 



This is a general expression for J?, the remainder after n + 1 

 terms of the expansion of F(a + h) by Taylor's Theorem. 



, For a particular case take /(a;) == (a^ a)^ 1 , where p, is any 

 positive number which is not less than q ; then all the con- 

 ditions with respect to f(x) are satisfied: and we have 



/(a + />)=#*', 

 and /^ (a + 0h) = (p + 1) p ... (^ - ^ + 1) (0^)". 



Hence : ' 



L (1 - 0)"-^ A" +1 F n+l (a + h6] ( 



= : -* ~ 



In the particular case in which p = q~we have from (3) 

 _(l-0) n - f h n + 1 F n+1 (a + 0h) .. 



If in (4) w^e put p = n we have Lagrange's form of the 

 remainder, which is given in Art. 92 ; if in (4) We put p = 

 \ we have Cauchy's fonn of the remainder, which is given 

 in Art. 110. 



Other particular forms may be readily obtained. Thus in 

 (3) pift # = ; then since [jO must be replaced by. unity we 

 have 



(l-e) 



