60 DIFFERENTIAL AND INTEGRAL CALCULUS. 



and x l ] (o? 2 suppose), which is a fortiori between 

 and h. But 



~ W -2 



}-V'(a}-xf"(a)-...-r?- -E, 



which also vanishes when # = 0; and the same again 

 applies, and so on, until we come to the result that 

 f* (x), or (j> n (a 4 x) E must vanish for some value of x 

 between and h. (This does not vanish when x = 0, so 

 far as we know ; and, therefore, the argument does no 

 longer apply). Thus E = < n (a -f 6Ji) where 9 is some 

 positive proper fraction, so that we have 



(f>(a + h) = (f>(a} + h<l>' (a) + <" (a) +...+ p (/>" (a + 0A), . 



being some positive proper fraction, provided that as 

 was assumed in the proof, </> (a?), and all its differential 

 coefficients are finite (i.e. not infinitely great, they may 

 vanish) and continuous between the limits a and a + h. 

 Hence, if we find the least and greatest values which < w (x) 

 can have between these limits we shall have certain limits 

 between which E must lie. 



Thus suppose </> (#) = V(^)j # = 1, ^ = Tcfu = '01, w = 3, 

 and we have 



1 i_ h IMV_zL 



1)^2 uoo; ' 4xl i 



Hence the sum of the first three terms will differ from the 

 truth by < 1gAAAAAA , or to eight decimal places 



IbUUUUUU 



**) = 1-f -005 - -0000125, or V(101) = 10'049875. 



