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■VII. On an Extension of the Principles of the Calculus of Varia- 

 tions. By Professor Challis, M.A., F.R.S., F.R.A.S* 



THE considerations I am about to offer relative to the prin- 

 ciples of the Calculus of Variations have arisen out of 

 attempts to solve the problem of the maximum solid of revolution 

 of given surface and given length of axis. That a solid exists, 

 the largest of all solids of revolution whose surfaces are of given 

 area and extend continuously from one extremity of the axis to the 

 other, there can, I think, be no reason to doubt ; and it will be a 

 reproach to mathematical science if it remains undiscovered. I 

 discussed this problem in the Philosophical Magazine for August 

 1861, and again in that for September 1862; and after full con- 

 sideration of the reasoning there employed, I do not see how, 

 according to the principles of the Calculus of Variations as usu- 

 ally understood, the line which by its revolution generates the 

 surface of the required solid can be any other than the one which 

 that reasoning gives, viz. a composite but continuous line con- 

 sisting of equal ordinates at the extremities of the axis, connected 

 by a portion of the curve generated by the focus of an hyperbola 

 rolling on a straight line. But, as I have remarked at the be- 

 ginning of another communication on the same problem in the 

 Philosophical Magazine for March of this year, the solid enclosed 

 by this surface does not possess the character of a maximum. 

 The aforesaid reasoning can therefore only be regarded as a 

 reductio ad absurdum, proving that the principles on which it 

 rests require to be corrected or supplemented. This I consider 

 that I have in effect done in the paper just cited; but as I find 

 that the principles which the new method involves may be more 

 distinctly stated, and the reasoning itself be put more logically 

 than I have there succeeded in doing, I propose now to resume 

 the subject. 



The argument may commence with assuming the usual expres- 

 sion for the function u that is to be a maximum or minimum, viz. 



u = B + \ (Sy —p8x)Adx, 



B being the part extricated from the sign of integration, and 

 containing the values of Bx and 8y at the limits, and A being a 

 given function of x, y, and the differential coefficients p, q, &c. 

 It may here be remarked that the substitution of x-\-hx and 

 y-\-hy for x and y in the calculus of variations for the purpose 

 of investigating rules for obtaining functions that satisfy the 

 conditions of maxima and minima, is analogous to substituting 

 in the differential calculus x + k and y-\-k for x and y in order 



* Communicated by the Author. 



