394 Prof. Challis on Integrating Differential 



putting Sforjoj 1— ~- ), that equation after due simplifications 



OPPOTY1PS 



Vy dx y* a y* \ay a y) v r ; J 



= f PE SSxyPMF 2? 3 p¥ 2P*S\ 

 U P / \«y d# «y 2 z/ 2 / 



This is the equation in the present instance which corresponds 

 to the equation ^r=0 in the general theorem. Now it is readily 

 seen that this equation is not verified either by F = or by 



j— = 0; but if both F = and — =0, it is reducible to 



S |3SP + P ! _2P1 

 L py a y J 

 which equation is satisfied if S = 0, and also if 



2V\a y) P \ 2a)' 

 The supposition that S=0 gives p = Q and y= -p-, which results 



(since p = is satisfied if y=0) accord with th,ose derivable from 

 the equation ($) when 6 2 =0. The other value of S is identical 

 with its assumed value ; which proves that the equation {tj) is 



satisfied by F = and — = 0. As the equation (rj) is only an- 

 other form of the equation (e), this result is confirmatory of the 

 inference already drawn from the latter. 



The forms of the function F assumed in the last two instances 

 occur in the processes for solving two well-known problems in 

 the Calculus of Variations. In Chapter IV., art. 57, of Mr. 

 Todhunter's e Researches in the Calculus of Variations' (an Essay 

 which obtained the Adams Prize in 1871), one of these problems 

 is thus enunciated: — "Required the plane curve joining two given 

 points which by revolving round a given axis in its plane will 

 generate a surface of minimum area." After referring to dis- 

 cussions of this problem by several previous writers, and stating 

 that the result given by the ordinary rules of the Calculus of 

 Variations is a catenary of which the axis of x is the directrix, 

 Mr. Todhunter proceeds to examine this solution. He first 

 admits, as established by previous researches, that " sometimes 

 two catenaries can be drawn, sometimes only one, and sometimes 

 none," and then finds, by carrying the investigation further, that 

 " when two catenaries can be drawn the upper corresponds to a 



