36 Prof. Challis on a new Integration of Differential 



we have shown that the equation (7) may be verified by means 



dA. 

 of the equation A = and its derived equation — =0. That 



integral may consequently be applied in the solution of the pro- 

 posed problem. 



What is gained by this process is, that an additional arbitrary 

 constant is available for satisfying the given conditions of the 

 problem. The integral of Ap = fails to give the continuous 

 solution because it contains only two arbitrary constants. The 

 required curve might either be described by first constructing 

 the evolute from its equation and then unwinding from it a 

 cord of arbitrary length, or by constructing the catenary given 

 by the equation Ap — Q and then drawing a curve distant from 

 the catenary at all points by the same arbitrary interval. As I 

 do not think it necessary to enter at present more into detail 

 respecting the application of the new integration to this instance, 

 I shall now proceed to apply it to another problem. 



Problem III. Required the maximum solid of revolution of 

 given superficies, the generating line of the surface being sup- 

 posed to join any two given points in a plane passing through 

 the axis of revolution. 



In the usual enunciation of this problem the two points are 

 supposed to be in the axis of revolution. But according to the 

 principle of integrating along consecutive points of the curve, as 

 stated in the Lemmas at the beginning of this communication, 

 the process for finding the required differential equation is the 

 same wherever the given points be situated, the coordinates of 

 their positions coming under consideration only in determining 

 the limits of the integration. This problem has hitherto not 

 been generally solved even for the case in which the given points 

 are on the axis, I propose to apply to it a treatment precisely 

 analogous to that employed in the preceding problem. 



We have in this example to find a relation between x and y 

 which shall satisfy the condition 



S/V ~ 2ays/T+f) dx = 0, 



— 2a being an arbitrary constant taken for convenience with a 

 negative sign. By the usual rules of the Calculus of Variations, 



A= ^vfe + 7T^> and (V-^)a=o. 



Hence A=0, and by consequence Ajd = 0. I shall not have oc- 

 casion to refer to the discontinuous solution which mathemati- 

 cians have elicited from the latter equation, my sola object being 

 to discover a solution by a continuous line joining the two points, 



