AS COMMONLY INVESTIGATED. 223 



which Poisson changes to 



F (x + h) = F (x) +h-.P; 

 but Avith what justice we shall now consider. 

 The development 



F (x + h) = F (x) + R (X, h) 

 {R (X, h) = 0} * 



h = 



IS merely an analytical expression for the hypothesis that h is independent of 

 X, and therefore unquestionable. But, without venturing so far as the suppo- 

 sition of R (x, h) containing a factor which is a power of h, what right have we 

 to assume that it contains a factor which is even a pure function of h? or, in 

 other words that 



R{x,h) = ^h).Q{x,h); 



It may be answered that in no other form than as a multiple could h destroy 



terms containing the independent variable x. But how much is here assumed 



_ 1 



will be perceived by considering that, in such simple functions as x ^, or log. 

 — — — , that vanish when h = 0, we do not immediately see that h^' enters as a 



factor; and although it might be demonstrated that h is so involved, yet what 

 are we to say in regard to the extension of the assumption to that host of tran- 

 scendental functions, the varying and almost capricious relations of which 

 analysts are only beginning to understand? 



Even in the simple vanishing function log x, we know that x* cannot be 

 said to enter as a factor, unless we admit that a is infinitely small; and other 

 cases might be found still more perplexing. 



A geometrical illustration will, however, suffice for this purpose. Thus, 

 representing graphically three functions of the form 



J — h* 

 {y= ^h =01 



h = 



{j = ^ (X, h) = 0} ; 



h = • 



the first is seen to be continuous and parabolic; and, consequently, if the two 

 latter partake of the same degree of continuity, we may conclude that a could 



* This notation denotes the function to vanish when h = 0. 



