278 Prof. Challis on a Problem in 



minium, covered as it is with a thin, insoluble, oxidized film, 

 becomes negatively excited in the immediate region by the hy- 

 drogen which is set free, whilst the film itself is preeminently 

 positive towards the metallic surface that it overlies, then the 

 phenomenon observed must undoubtedly make its appearance." 

 I believe that my experiments have justified this assumption. 



XXXV. On a Problem in the Calculus of Variations, in reply to 

 Mr. Todhunter. By Professor Challis, M.A., F.R.S., 

 F.R.A.S.* 



I HAVE only a few concluding words to say respecting the 

 problem in the Calculus of Variations, the consideration of 

 which is resumed by Mr. Todhunter in the September Number, 

 the discussion of it having advanced to a stage at which, so far at 

 least as my views are concerned, a definite issue may be stated. 

 I have all along understood the problem to be that which Mr. 

 Todhunter has thus enunciated : To determine the greatest solid 

 of revolution, the surface of which is given, and which cuts the 

 axis at two fixed points. The generating line of the surface I 

 suppose to be subject also to the condition of being continuous, 

 so far as not to change its direction per saltum. It has been 

 shown in the course of this discussion that different lines fulfil- 

 ling this condition, together with that of cutting the axis at the 

 two points, give solids having the same superficies but different 

 degrees of magnitude. As these gradations cannot go on unli- 

 mitedly, there must be a limiting or maximum solid the surface 

 of which fulfils the same conditions. It would be possible by 

 practical tentative methods, although it would be an unscientific 

 proceeding, to mould a given quantity of matter into the required 

 form. The maximum, therefore, exists in rerum naturd; and to 

 the discovery of its form by analysis all my efforts have been di- 

 rected. I cannot recognize as having any claim to be called the 

 solution of the problem a form which does not fulfil the above 

 conditions. 



In the July Number I have given two investigations — one de- 

 pending on the equation Ap = 0, and the other on A = 0. I 

 showed at the same time that the former equation could only 

 give a line symmetrical with respect to the axis of revolution, 

 and consequently, if any, only a conditional maximum, and that 

 the absolute maximum must be deduced from the equation A = 0. 

 Further, it was inferred from the radical involved in A (assuming 

 always that the function or functions to be found admit of exact 

 algebraic expression), that the equation A=0 should be resolved 



* Communicated by the Author. 



