196 Prof. Challis on the Principle of Discontinuity in 



sory but identical. It is true that the equation (e^, not (e'j), is, 

 or ought to be, identical in the case under discussion. But what 

 of that ? I can discover no semblance of solidity in his remark. 

 Doubtless nothing better can be advanced on his side of the 

 question. For (e'j) cannot be expunged. 



With respect to what he says of the six equations and the six 

 quantities to be determined, I would merely remark that, when v 

 can be expressed as a rational function of u, we ought, in general, 

 to be able to obtain an analogous expression for u in terms of v. 

 The method is one of verification in ordinary cases. 



8. It is clear, from what has been said, that if v, instead of 

 being equal to u b } be a rational function of u defined by 



v = a 5 + a 4 u + a 3 u 2 + . . + a u 5 , 



Lagrange's theory will be inapplicable in virtue of the equation 



0=/* 5 — a 6 + (/* 4 — a 4 )w + (fi 8 — a 3 )w 2 + . . + (fi — a )w 5 . 



We cannot superadd a second rational relation to the first. 



What mathematician, who is not urged on by his fear of the 

 algebraical resolution of equations, can doubt this ? 



August 1862. 



XXVIII. On the Principle of Discontinuity in Solutions of Problems 

 in the Calculus of Variations. By Professor Challis, F.R.S.* 



THE following communication was written in consequence of 

 my receiving, from Professor Lindelof of Helsingfors, a 

 copy of the work, recently published by him in conjunction with 

 M. PAbbe Moigno, entitled Leqons de Calcul des Variations. I 

 believe that I owe this mark of courtesy (for which I beg to take the 

 present opportunity of expressing my acknowledgement) to the 

 circumstance of my having given, in the Number of the Philoso- 

 phical Magazine for August 1861, a solution of the following 

 problem : — To determine the surface of revolution of given su- 

 perficial area, and passing through two given points of the axis, 

 which shall include the greatest volume. The solution there 

 proposed is adverted to in p. 224 of the work in these terms : — 

 "On their part, MM. Challis and Todhunter, admitting that the 

 line which is to unite the points A and B is composed, first, of 

 two straight lines raised perpendicularly to the axis at the points 

 A and B, then of a curve joining the extremities of these straight 

 lines, equally arrive at giving a discontinuous curve as the solu- 

 tion of the problem in question." Respecting this statement, I 

 beg permission to make the remark that it takes no notice of an 

 important difference between Mr. Todhunter's treatment of the 



* Communicated by the Author. 



