the Calculus of Variations, 203 



one value of A and exclude the other would be nothing short of 

 error." 



I admit of course that the equation A'— A" = must be satis- 

 fied ; but I do not admit that this is the one equation which the 

 form of the curve is required to satisfy. 



We have to satify both A' = and A'=0. We may of course 

 try to assist ourselves by discussing the equation A'— A" = 0; 

 but any result which we deduce from the last equation will not 

 be applicable to the problem we are solving, unless it makes both 

 A! = and A"=0. It is therefore unnecessary to examine the 

 validity of the process which is applied to the equation A'— A" = 0, 

 so long as it is obvious that the result does not make both A' = 

 and A" = 0. 



That the equations A' = and A" = are not satisfied when 

 we take for the required curve a segment of a circle, can be im- 

 mediately ascertained by trial. 



Or we may establish this assertion by referring the curve to 

 polar coordinates. In this case we shall have only one value of 

 r, corresponding to one value of 6 ; so that we have no occasion 

 to consider two values of A. The differential equation in polar 

 coordinates, as given in the Magazine for March, is 



7*sin0 (r + r") r'cos#— Sr sin 6 _ sin 6 



where accents denote differential coefficients. Now the equation 

 corresponding to a segment of a circle is r = Cj cos (0 — C 2 ), 

 where C x and C 2 are constants. It will be found immediately, 

 on trial, that so long as C 2 is not zero the equation is not satis- 

 fied, whatever sign we give to the radical. 



Thus it follows that the method which I am examining is op- 

 posed to the fundamental principles of the Calculus of Variations. 



I wish to advert to one of the results enunciated in March, 

 because I am uncertain whether it is still maintained, or is aban- 

 doned as erroneous. The result was enunciated thus: — "The 

 solid consisting of a cylinder and two hemispherical ends of the 

 same radius, is larger than any other solid of revolution having 

 the same amount of surface and the same length of axis/'' 



I urged two objections against this, one of them being that 

 the fundamental equation A = is not satisfied. I cannot agree 

 with Professor Challis that he sufficiently meets the objection. 

 It seems to me that there are only three ways in which the ob- 

 jection may be combated : (1) by showing that the equation 

 A=0 is satisfied; (2) by denying that it is necessary to satisfy 

 the equation A = 0; (3) by showing that although it is in 

 general necessary to satisfy the equation A-=0, yet there are 

 special reasons which remove the necessity in the present case. 



