DIFFERENTIAL AND INTEGRAL CALCULUS. 61 



6 will in general be a very complicated function both of 

 a and h, and it would be a very difficult matter to determine 

 it ; but the important property of it is that it must always 

 lie between and 1. 



If <f> (x) be a rational algebraical function of the n + 1 th 



degree, 6 = , which appears to be the only case in 



which it is constant. If < (x) be e s , 6 is a function of h 

 only, given by the equation 



or v ,- lo 



a function of li only. 



If $ (x) be x 5 and n = 3, we have 



(a + h) 5 = a 5 + k 5a 4 + - 20a 3 -f - 60 (a + 6A)* 



= a 5 + 5a 4 A -f 10a 3 /* 2 + 10a 2 /* 3 + 5a/i 4 + A 5 , 

 so that (a + Oh)* = a 2 + \ah -f y 1 ^, 



a //a 2 la IN 

 or0 = T + A / i + F H" 7 : ) i taking the + sign since 



A V ^ 2 " 1 ^/ 

 ^ is positive 



_ /I a I \ a //a 2 a 1 \ 



" \2 A 10/ ' A <y vA" 2A lO/ " 



The greatest and least possible values of 6 are found 

 by putting h = co , h = 0, and are -TTTTV and - . Thus 



] (a 1\ 2 a 2 I a 1 



so 



