204 On a Problem in the Calculus of Variations. 



But the objection is not removed by reasoning which does not 

 bear on these points : it is not removed, for example, by showing 

 that the proposed solution satisfies A = at some points, and 

 satisfies p — at all the other points. 



It is almost superfluous to advance another argument against 

 the untenable result ; but I may just mention that, by changing 

 the hemispherical ends into semispheroids, we can form a solid 

 having the same surface and the same length of axis as that 

 which is erroneously called the greatest, but having a greater 

 volume. 



Having shown by examples and by theory that the results 

 given in the Numbers for March and for July are inadmissible, 

 I shall proceed in the third place to offer some remarks as to the 

 possibility of solving the problem with the condition of continuity. 



The word continuous may have more than one meaning ; but I 

 think that the following remarks will apply with any meaning 

 which is likely to be assigned. 



The figure which by its revolution round the axis generates 

 the solid of greatest volume with a given surface, is a figure 

 formed of an arc of a semicircle and a straight line which coin- 

 cides in direction with the bounding diameter. This figure will 

 be regarded as non-continuous by those who seek for a conti- 

 nuous solution. But we know that we can in general draw a 

 continuous curve through any assigned number of points, how- 

 ever large. Hence we can in effect make a continuous curve 

 coincide as nearly as we please with the non- continuous curve 

 which gives the greatest solid. The best method of conceiving 

 this to be done is to employ the theorems which serve as the 

 foundation for the expansion of functions in terms of sines and 

 cosines of multiple angles. 



It seems to follow from this consideration that it is in vain to 

 seek for any solution, continuous or non-continuous, which dif- 

 fers from that determined by the semicircle and straight line. 



Again, whether a solution be continuous or non- continuous, it 

 must satisfy the fundamental equation of the Calculus of Varia- 

 tions which I have denoted by A = 0; and it does not seem pos- 

 sible to satisfy this equation except in the manner indicated iu 

 the Magazine for June. 



The very interesting investigation respecting the course of a 

 ship, to which Professor Challis refers, was unknown to me 

 when I published my ' History of the Calculus of Variations/ I 

 regret this, because the subject of discontinuous solutions of 

 problems in the Calculus of A r ariations appears to me important ; 

 I have given several examples, and I should have been glad to 

 have included in my work a notice of every case which had been 

 discussed. I venture to suggest, without, however, laying much 



