380 Proceedings of the Royal Irish Academy. 



(h) It must also be small in the sense that, when our only object 

 is to determine the sigji of an expression, a term involving it may be 

 neglected in comparison with a term involving only quantities of 

 arbitrary and finite magnitude, or that one involving only quadratic 

 or higher powers of the variation may be neglected in comparison 

 with, one involving only linear terms. "We may therefore vsrite 



h/r,s,...) <]c^ 



where k is small. 



(c) Any variation typified by Syf"- ^> • • •) may, at any point of the 

 integration, have any arbitrary finite value consistent ivith (i). The 

 necessity of imposing this restriction is evident ; for, if the fluxion 



had a finite and positive value of arbitrary magnitude over a finite 

 range of integration with regard to dx, it is evident that 



must also, at some points of the integration, have a finite value of 

 arbitrary magnitude, contrary to {h). 



I 6. When there is but one independent variable x (the suffix being 

 omitted), the conditions of § 2 will be sufficiently demonstrated by 

 taking two dependent variables, yi and y^ ; for it will be evident, from 

 the method of proof, that the conditions for any number of dependent 

 variables can be established in a similar way. 



For convenience of explanation, let the stationary values of y^ and 

 2/2 which correspond to the stationary 

 solution be represented by the ordinates 

 of a curve OP, and let x be the abscissa. 

 The curve will, of course, be in three- 

 dimensional space ; but it is not neces- 

 sary to draw the axes of reference. 

 Let the limiting points be 0, P. 



Let . . . AB ... P be the broken solution with which we are 

 to compare it, AB being a continuous element^ of the curve of 

 length Dx. (The figure is drawn for the case where the tangent may 

 be discontinuous in direction.) 



1 By an element is meant a quantity whose square may be neglected in comparison 

 with its first power. For our purpose, therefore, we mean by an element a length 

 Dx less than k. 



