DIFFERENTIAL AND INTEGRAL CALCULUS. 59 



In this particular case, we know from ordinary algebra, 



jn 



the remainder of the series after n terms, viz. - - 7-. , 



x (x - h) ' 



and see that the series is convergent only when is nume- 



c 



rically < 1, so that the remainder may be made as small as 

 we please by taking a sufficient number of terms. It is 

 then advisable in general to determine limits for the re- 

 mainder after n terms of this expansion, and slightly 

 altering the notation, we assume 



$ (a + h) = $ (a) + h# (a) + </>" (a) +... 



and seek to determine the form of R in some way. Now 

 denote the .following function of x 



$ (a + *)-<!> (a) - xf (a) - * () -^S, . 



by F(x}j then by equation (A), we have F(h)=Q. Also 

 F(Q) = <l>(a)- 0(a) = 0. Hence, F(x] vanishes for the 

 two particular values of x, 0, and h. But if a function 

 of x vanish for two particular 1 values of x and do not 

 become infinite between those limits, then since it cannot 

 be always increasing or always decreasing, it must at some 

 point change from increasing to decreasing or the reverse, 

 i.e. its differential coefficient must change sign, which not 

 being infinite, it cannot do without passing through the 

 value 0. Hence F' (x) vanishes for some value of x 

 between and h (x t suppose). But 



and therefore vanishes when x 0. Hence, the same 

 argument applies to F 1 (a;), and its differential coefficient 

 or F" (x) must vanish for some value of x between 



