282 On a Problem in the Calculus of Variations, 



proximate integrations, but I do not apprehend that they would 

 be insuperable. 



For the sake of further illustration of the principles of the 

 Calculus of Variations, I propose to conclude with some remarks 

 on a problem similar to the one that has been under discussion, 

 but not to be confounded with it. Let it be required to find 

 the greatest solid of revolution of given superficies, the genera- 

 ting line of the surface consisting of ordinates perpendicular to 

 the axis at two given points, and of a curve connecting them. 

 Mr. Todhunter proposed this problem in p. 410 of his ' History 

 of the Calculus of Variations/ and in all essential points effected 

 the solution of it by proving that the curve must join on conti- 

 nuously with the ordinates, and that the ordinates must be equal. 

 In the Philosophical Magazine for August 1861, I found that 

 the relation between the ordinate y and the length s of the curve 

 could be thus expressed : 



a**)-* 



2a being what I have hitherto called X, and k and b arbitrary 

 constants introduced by the integrations. MM. Delaunay and 

 Sturm had proved [Journal de Liouville, vol. vi. p. 315) that the 

 curve to which this equation belongs may be described by the 

 focus of an hyperbola rolling on a straight line, the latter ma- 

 thematician discussing it with reference to the problem treated 

 of in the previous part of this communication. But what I chiefly 

 wish to remark is, that although the above equation was obtained 

 by the integration of the equation Aj9 = 0, it satisfies the equa- 

 tion A=0 — that is, the equation 



d ayp 



2/+av / 1+/ __- 7 . TT ^=o ) 



as might easily be verified. Hence the factor p has no such sig- 

 nificance here as in the former problem, and the curved line is 

 altogether continuous. In the March Number of this year, I 

 have called in question the truth of this solution ; but as this 

 was done in consequence of the error I had fallen into relative to 

 the integral of the equation A=0, as already pointed out, I beg 

 to withdraw what I have there said. I see now no reason to 

 doubt that the form of the curve above mentioned is the true 

 solution, although it might be difficult to apply a test. For the 

 particular case in which b = 0, and therefore ^ — 0, the curve is a 

 semicircle terminating at the two points j and thus this result 

 coincides with that restricted solution of the former problem 

 which was deduced from the equation A = 0, as it manifestly 



