the Calculus of Variations. 281 



tion would be compatible with a definite algebraic solution, 

 which is contrary to what was before proved. Thus, under the 

 circumstances of this problem, Ap = and A = are not equiva- 

 lent equations, only the former admitting of an exact form of so- 

 lution ; on which account I still think I am right in calling the 

 solution a conditional maximum, the condition in this instance 

 being that the line be symmetrical with respect to the axis of x. 

 There can be no question that a maximum is in this manner 

 really obtained, if only for the reason that of all solids, regular 

 or irregular, of given surface, the sphere is the greatest; but evi- 

 dently the maximum we are in search of must not be subjected 

 to that condition. It may also be noticed that the solution, 

 since it gives a discontinuous line, is analogous to that of the 

 problem of the shortest course of a ship, and confirmatory of the 

 principle that the Calculus of Variations is comprehensive of dis- 

 continuous solutions. 



I can see no good reason for the attempt (in the September 

 Number) to show that " a figure formed of an arc of a semicircle 

 and a straight line which coincides in direction with the bound- 

 ing diameter " may be regarded as a continuous figure, nor do I 

 perceive what bearing this idea has on the treatment of the pro- 

 blem. What objection can there be to the common-sense view 

 that such a figure is discontinuous ? Also I cannot forbear noti- 

 cing what strikes me as an inconsistency, where, at the close of 

 his communication, Mr. Todhunter first expresses the opinion 

 that the use of the equation Ap = may be dispensed with, and 

 then states that he accepts, as the only true solution of the 

 problem, one which is actually derived from the equation Ap — O, 

 and is not derivable from the equation A = 0. It cannot but be 

 a right rule of analysis, to regard as non- equivalent expressions 

 symbolically distinguished. 



The following are the conclusions I have come to from this 

 reconsideration of the question : — 



(1) The Calculus of Variations furnishes no other solution in 

 definite algebraic form than that which gives for the generating 

 line of the surface a discontinuous line, consisting partly of a 

 semicircle and partly cf a straight line coincident with the axis. 

 But this solution does not satisfy the conditions of cutting the 

 axis at the given points, and of continuity of form. 



(2) In order to satisfy those conditions, it is necessary to inte- 

 grate the equation A = by successive approximations; which 

 being done, the value of y is obtained in a series which is not 

 the development of a definite algebraic function of x. 



The form of the surface might thus be obtained for different 

 sets of numerical values of the constants. I do not know what 

 difficulties would be met with in the attempt to effect such ap- 



