68 DIFFERENTIAL AND INTEGRAL CALCULUS. 



all the other terms cancelling. Hence 



as before, or writing X = x + h, we have 

 <fr(x + h) = <l>(x) + 11$ (x) + ^ p (x) +-..+? <#>* 



If <' (a;) be a function which either increases with x or 

 decreases as x increases, since 



(a? + A) = (x) -f- kp (x + 0A), 

 we have, putting h = 1, 



0(^+1) -((*) = <' (a;+0), 



which, under the circumstances supposed, always lies be? 

 tween <f>' (x) and cf>'(x }-!). Thus if <j>(x] be sec' 1 (a;), 

 and therefore 



' 



we have 

 sec' 1 (z+1)- sec" 1 (a;) > 



X 



or if we take x = n, n- l,n 2, ... 1, successively, we have 



1_ 1 



* ' (n + 1) VO 



sec'Sz sec'Vn 1) > - 



w 3 V(8) 2 V(3) ' 



1 



(1)> 2V(3)' 

 the other limit not applying here (though quite true) ; 



therefore 

 sec" 



