54 On an Extension of the Principles of the Calculus of Variations. 



at its extremities with the given points by two straight lines in- 

 clined at small angles to the axis PQ. It is evident that by- 

 varying the angles of inclination, solids of different degrees of 

 magnitude will be obtained, and so much the greater as those 

 angles are less, the maximum of all being that for which the 

 angles of inclination are indefinitely small. This maximum is 

 the same as that which results when the equation y — is. taken 

 into account, which Mr. Todhunter regards as the only solution 

 which our problem admits of. It satisfies the equation Ap = 0, 

 but not the equation A = 0. I assent to the opinion expressed 

 by M. Lindelof (Leqons de Calcul des Variations, p. 225), that 

 this is not properly a solution of the proposed problem. I can- 

 not perceive that it differs essentially from the solution of the 

 unconditioned problem of the maximum solid of given surface ; 

 and, at least, it does not possess the same interest as the deter- 

 mination of the form of the maximum solid which has a conti- 

 nuous surface passing through the given points. 



If I have at length succeeded in effecting a solution of this per- 

 plexing problem, I have at the same time pointed out the rea- 

 sons of the failure of previous attempts. One reason appears to 

 have been an imperfect knowledge of the rules of application of 

 the integrals of differential equations whose degree is higher than 

 the first. Instances of such application have been rarely handled 

 by analysts, and the principle of discontinuity which they gene- 

 rally involve is hardly yet fully admitted. As far as I am aware, 

 the first case of an integral of this class in the calculus of varia- 

 tions leading to a discontinuous solution is that which occurs in 

 the solution of the problem of the shortest course of a ship, 

 which is given in the September Number of 1862 (p. 197), but 

 was originally published so long since as 1834 in vol. i. of the 

 Philosophical Magazine for that year (p. 33). But the chief 

 cause of failure, which, however, is connected with the other, was 

 the confounding of two differential equations, A = and A/> = 0, 

 of different degrees, and regarding them as identical. On the 

 general principles of analysis it might be argued that, while there 

 are cases in which the equations lead to identical results, the 

 very fact of there being two equations is evidence that there are 

 also cases in which the results are not identical. For no sym- 

 bolical difference can be entirely without meaning. The fore- 

 going investigation has, I think, shown that in the present in- 

 stance one equation points to a relative or conditional maximum, 

 and the other to an absolute maximum. As I am not aware 

 that the two equations have been similarly treated in any previous 

 researches, I have ventured to call the use I have made of them 

 an extension of the principles of the Calculus of Variations. 

 Cambridge, June -16, 1866. 



