320 Mr MOORE, ON A DIFFICULTY IN ANALYSIS 



, ^ (.r) 1 ,-, , , i, • fix) 1 (a") 

 equal to ^ ' . . JNow by hvpotnesis ^ and ~ — - 



toAvards a finite limit as x converges indefinitely towards zero, whilst 



,„ _ „^ converges towards - or zero at the same time, according as >«, 



is greater or less than m. Therefore a quantity converging towards 

 a finite limit can throughout be equal to another which converges 



towards ^ or zero, which is absurd. Therefore J'{x) cannot be equal to 



(p (.r) for the general value of .r. q. e. d. 



CoK. If lim,=|j {J'{x)} is an infinitesimal of the »«"■ order, and 

 l\m,,^„{(p{x)], lim.,=o{0i(a;)}, lim ,=„ f02(.i')i. lim.„„{0,„_i(a;)} infinitesimals 

 of the ?«,"', ?«2^ ^m-\ orders, /"(x) cannot be equal to 



A(p{x) + B(p,{x) + C(p,{x) 

 for the general value of x. 



SCHOLIUM. 



Hence we see that the law of homogeneity, which is so essential an 

 element of all analytical developements, holds good at the limits of the 

 functions and variables as well as for values varying between finite 

 limits. We shall now see that this law is alone sufficient to demon- 

 .strate La Grange's principle within the proper limits of its application, 

 as well as to indicate at once the cases in which it is necessarily in- 

 applicable. 



3. I shall now enunciate and demonstrate La Grange's principle. 



THEOREM. 



If y(.r) be a function of x continuous between the limits and a;,, 

 and if lim^=o {y^W} is an infinitesimal of a finite and positive order oc, 

 the function J'(x) for any value of x within those limits may be analy- 

 tically represented by a series of terms of the forms Ax" + Bx^+ Cx^ + ^c. 

 where A, B, C are finite coefficients, and a, /3, 7 finite and positive 

 exponents. 



