404- MISCELLANEOUS PROPOSITIONS. 



373. We have already given two forms for the remainder 

 after n + 1 terms of an expansion by Taylor's Theorem ; see 

 Arts. 93 and 1 10 : these two forms, and others, may be 

 deduced from one general expression which we will now 

 investigate. 



Let <J> (x) and ty (x} be two functions of x which remain 

 continuous, as also their differential coefficients between the 

 values a and a + h of the variable x ; suppose also that be- 

 tween these values the differential coefficient ^r'(x] does not 

 vanish : then by Art. 98 



<f>(a+h)-<f>(a) _ (j>'(a -4- 0h) ,. 



<lr (a + h) - f (a) ~ ' " 



where is some proper fraction. 

 Denote by (f> (x} the function 



and denote by -/r (x) the function 

 /(a + h) -f(x) -(a + h- x)f'(x) - ... - 



We assume that F(x] and all its differential coefficients 

 up to F n+l (x) inclusive are continuous while x lies between 

 the values a and a + h ; as also f (x) and all its differential 

 coefficients up to f** 1 (x) inclusive : moreover we assume that 

 f 1 ** 1 (x) does not vanish between these values. 



Now x= 



and 



also $ (a + h} = 0, and ^ (a + h) = : 



thus we have from (1) 



