426 Mr. I. Todhunter on a Problem in the Calculus of Variations. 



where b is another constant, which is introduced by the inte- 

 gration. 



Since the curve is to meet the axis, we have y = at certain 

 points; hence 6 = 0, and the equation becomes 



{vwr^ +y }y=°- 



2a 

 Thus we have either —771 o, +y = 0, or y = 0. The first cor- 



4/(1 -f;? 2 ) 



responds to a sphere, and the second to a cylinder, of indefinitely 

 small radius. This solution is due to the Astronomer Royal ; 

 it is given in the Philosophical Magazine for July 1861, and 

 illustrated by figures. 



There is, however, one difficulty which still exists. On pro- 

 ceeding to verify the solution, we find that the terms denoted by 

 B vanish at the limits ; also A vanishes for all that part of the 

 solution which corresponds to the sphere ; but A does not vanish 

 for the remaining part of the solution. In fact, if we put y = 0, 

 we find that A = 2a. Thus instead of having 8w = 0, we have 



8u = 2a§ (hy—phoc)dx 



— 2d i \hydx } 



since p = when y — 0. The integration is to extend throughout 

 the range which corresponds to the cylindrical portion of the 

 solution. 



It appears to me that the following is the explanation of the 

 difficulty. It is seen on examination that a is a negative 

 quantity. Also by the nature of the problem, corresponding to 

 2/ = 0, the value of Sy must be positive. Thus 8u, instead of 

 being zero, is essentially a small negative quantity ; this, how- 

 ever, ensures that u is a maximum, and therefore we obtain all 

 that is required. 



The following consideration will, I think, show that it is in 

 vain to seek for any other solution instead of that proposed by 

 the Astronomer Royal. It is an admitted result that among all 

 figures of given surface the sphere is that which has the greatest 

 volume. Now, in the solution of the problem which we are con- 

 sidering, we obtain a figure the surface and volume of which 

 differ infinitesimally from those of a sphere ; and hence we con- 

 clude that the volume we obtain is greater than that of any pro- 

 posed figure which differs to a finite extent from a sphere with 

 the given surface. 



It should be observed that there are two forms in which the 

 problem may be proposed : we may suppose that the solid is 

 restricted to lie between two planes which are at right angles to 



