DIFFERENTIAL AND INTEGRAL CALCULUS. 67 



n being a positive proper fraction, 



and $ (x + 6k) = <f>' (x) + Ohf (x + mh), 



m being a positive fraction < 6 therefore 



- <" (x + nh) = QW $" (x + mh), 

 2 



" nh) 1 



- , ; = - when h = 0. 

 mft 2 



(This indicates that in any curve P$ if the tangent at R 

 be parallel to PQ, then when PQ moves parallel to itself 

 up to R, R ultimately bisects the arc PQ). 



Another proof of the theorem on the limits of the re- 

 mainder after n terms is somewhat shorter than Homersham 

 Cox's, which is the one we have given. This is as follows : 

 Assume 

 <f, (X) = $ (x) + (X- x) f (*) + &=f y {x} + _ 



and let F (z) = <f> ( X ) - <t> (z) - ( X - z) <}>' (z) 



(X-*Y 

 -\j~* 



then F(x) by reason of equation (J.), and 



and since F(z] vanishes for the two values x and Jf, its 

 differential coefficient must vanish for some intermediate 

 value {say x + 6 (X- x)}, but 



F (*) = - f (*) 4 *' () - (X- z] <j," (,) + (X- z) V (a) 



