Problems in the Calculus of Variations. 303 



said, I find that no argument is adduced which does not rest on 

 the assumption that the equations Ap = and A = are necessa- 

 rily identical. To this assumption, which is made without the 

 support of any reasoning, I oppose the general argument, that 

 as the two equations differ symbolically (the former being of a 

 degree superior by one to that of the other), they cannot be ne- 

 cessarily equivalent, inasmuch as in that case the symbolic dif- 

 ference between them would have no signification, which is con- 

 tradictory to the principles of analysis. This a priori reason I 

 proceed to confirm by the following particular considerations. 



There are instances in which Ap = and A = are both in- 

 tegrable per se, and the two integrals are identical. The Pro- 

 blem L, solved in my article in the July Number, presents one 

 such instance. In other cases, as that of the problem proposed 

 by Mr. Todhunter in p. 410 of his ' History of the Calculus of 

 Variations/ only Ajo = is integrable perse, but the integral 

 satisfies A = 0. In these two classes of solutions the Calculus 

 gives for each problem a unique result. 



There are also instances in which a solution is effected by an 

 integral of Ap = Q which does not satisfy A = 0. This is the 

 case with respect to the problem of the greatest solid of revolu- 

 tion of given superficial area, the surface being subject to the- 

 condition of passing through two given points of the axis. The 

 discontinuous solution of this problem obtained by Mr. Airy in 

 the Number of the Philosophical Magazine for July 1861, is 

 deduced exclusively from the equation Ap = 0. In the Num- 

 ber for June 1866 Mr. Todhunter has completed this solution 

 by proving that it actually gives a maximum ; but in so doing 

 he has expressly excluded the equation A = 0. In short, in this 

 instance the equation Ap = is treated as if it were independent 

 of the equation A = 0. I am entitled, I believe, to say that 

 Professor Cayley assents, as I do, to the solution in question. 

 Why, then, does he object to treating A = independently of 

 Ap = ? The result which he accepts proves that the two equa- 

 tions are not necessarily equivalent. 



In the July Number I have deduced solutions of Problems II. 

 and III. by an independent treatment of the equation A=0, 

 and have thus obtained, in the case of Problem III., the conti- 

 nuous solution of the problem of which, as stated above, the 

 Astronomer Royal gave a discontinuous solution. The novelty 

 of the process I have adopted consists in deriving from the equa- 

 tion A = 0, regarded as the differential equation of a curve, the 

 equation of the corresponding evolute, and then employing one 

 of the involutes to satisfy the conditions of the proposed problem. 

 In the case of Problem II. the equation of the evolute was ex- 

 plicitly obtained, and the appropriate involute could consequently 



