the Calculus of Variations. 279 



into A'=0 and A" = 0, to meet the case of two ordinates to the 

 same abscissa. The equation Ap=0 did not require the same 

 resolution, on account of its fulfilling the above-mentioned con- 

 dition of symmetry. Having on these principles obtained, by 

 processes which will be presently again adverted to, two solids 

 of definite forms, 1 proceeded to apply tests to ascertain whether 

 they were really maxima. It is well known that results given 

 by the rules for finding maxima and minima must satisfy certain 

 criteria before they can be pronounced to be solutions of the 

 proposed questions. In default of general criteria, it is neces- 

 sary in the Calculus of Variations to have recourse to tests by 

 trial of particular instances. The tests I applied happened to 

 accord with the supposition of maxima ; and I concluded, it ap- 

 pears on insufficient grounds, that such was the character of the 

 two forms. Mr. Todhunter, by a more skilful application of 

 tests, has proved (in the September Number) that the forms in 

 question are not maxima. Of course I admit without reserva- 

 tion that all the inferences I drew from the supposed maxima 

 fall to the ground. By availing myself of this correction of my 

 mistakes I have been enabled to arrive at some definite conclu- 

 sions respecting the proper process for solving the problem, to 

 which I shall direct attention after giving a few preliminary ex- 

 planations. 



Relative to the solution drawn from the equation A = 0, Mr. 

 Todhunter urges that "we have to satisfy both A' = and 

 A" = 0." This I conceive that I have done. I have employed 

 integrals of these equations which may be written \ y'd% = M! + O 

 and f y"dx = ~M" + C", the former of which satisfies A' = 0, and 

 the other A" = 0, as is evident by simply differentiating them. 

 Also the equations are thus satisfied by general or indefinite in- 

 tegrals, which is an indispensable condition. I next proceed to 

 use these values of \y' dec and \y" dec strictly according to the in- 

 dications of the analysis, always taking into account that y 1 and 

 y" are ordinates to the same abscissa, and I obtain definite inte- 

 grals between the values^ = infinity and p= 0. The result is 

 that the area cut off by the larger ordinate is equal to the length 

 of the bounding curve multiplied by the constant \. As that 

 ordinate cuts the curve at right angles, this result indicates that 

 the curve is a circle of radius X. Every step of this process is 

 legitimate if the next preceding step is legitimate ; and therefore 

 the legitimacy of the result depends on the equations A' = and 

 A"=0 having been both satisfied. All this argument, how- 

 ever, presupposes that the equation of the curve admits of defi- 

 nite algebraic expression, the resolution itself into two equations 

 being an operation performed on algebraic principles. 



