218 Prof. Challis on the Solution of a Problem in the 



terms of the H or G series vanish simultaneously, either with or 

 without the corresponding terms of the / series vanishing too : 

 this inquiry, which is necessarily tedious, and the result of which 

 it is easy to anticipate, must be adjourned to a more suitable oc- 

 casion. 



If we call A the new superior limit, we have found j 



A-v = <£, 



where, $,is the collective number of values of x included between 

 a and b for which any function G r x vanishes, whilst II,._ 1( 2? and 

 H r+1 a? have like signs; but since, when G r x = 0,f r _ l % and/ r+l # 

 must have like signs, <f> may be denned more simply as the num- 

 ber of values of x between the given limits for which simulta- 

 neously, and for any value of r, G r x vanishes whilst G r _!^r and 

 G r+l x have like signs. 



This quantity <f>, the difference between the limit and the 

 number of roots limited, may be odd or even, and not necessa- 

 rily the latter, as is the case in all existing theorems of a similar 

 nature. 



P + P 

 Since A = *—— — , it follows that when P = 0, i.e. whenever 

 Z 



the passage from a to b leaves the number of permanences in 



the H series unaltered, the limit p given by Fourier's theorem 



may be replaced by ^ or ~ — 1, according as p is or is not divi- 

 sible by 4. 



XXXIV. The Solution of a Problem in the Calculus of Variations 

 by a New Method. By Professor Challis, M.A., F.R.S., 

 F.R.A.S.* 



THE problem to which the following investigation more 

 immediately relates is the same as that discussed in the 

 Numbers of the Philosophical Magazine for August 1861 and 

 September 1862, viz. to determine the maximum solid of revo- 

 lution, the surface of which is of given area and cuts the axis in 

 two given points. In the second of these communications I 

 have argued that there must be a particular solid which satisfies 

 the proposed conditions, because a surface of revolution of given 

 superficies might be such as to pass through the given points 

 and at the same time enclose as small a volume as we please, or 

 any volume that we please below a certain magnitude. Also I 

 maintained that there is no valid reason for concluding that the 

 Calculus of Variations fails to indicate this maximum, and that 

 to effect the solution of the problem it is only required to dis- 

 cover the proper process of investigation. Having recently had 

 * Communicated by the Author. 



