200 The Principle of Discontinuity in the Calculus of Variations, 



Such objection is, I think, completely removed by the reasoning 

 applied to the former problem. Consequently, the discontinuity 

 being for these reasons admitted, it will be necessary to obtain 

 a new expression for the differential of the surface. That is, 

 under the sign of integration there will be the additional terms 

 ardr -J- ar^dr 1 , r and ? J being the distances of points of the circular 

 areas from A and B respectively, and the integrations being 

 taken from r = to r=y , and fromr' = to 7 J =y v The quan- 

 tity outside the sign of integration to be equated to zero will 

 thus be 



and as y and y l are not given, we shall have 



/i ■ c, + 1=0, and — jz — — - —1=0. 

 a/1+V Vl+Po 2 



Hence it follows that p —-)-cc and jo, = — oo, and that the 

 extreme ordinates are tangents to the curve. The original dif- 

 ferential equation shows, by putting p = co , that the values of 

 these ordinates are determined by the equation 2/ 2 =#. It has 

 been found by M. Delaunay, as stated by M. Lindelof in p. 220, 

 that the curve might be described by one of the foci of an hyper- 

 bola rolling upon the axis of x. A line so described would be 

 a closed curve ; but according to the foregoing reasoning, only 

 the portion concave to the axis is applicable to the proposed 

 question. 



The problem has thus received a unique and definite answer 

 by a process each step of which is necessarily true if the prin- 

 ciple of discontinuity be admissible. The reasoning has also 

 shown that the assumption usually made in the treatment of this 

 problem, that 6 = 0, is not allowable, and that it simply has the 

 effect of making an answer to the question impossible. Also I 

 am unable, for the following reasons, to assent to the considera- 

 tions in pp. 224 and 225, by which M. Lindelof endeavours -to 

 show that the problem does not admit of a solution. It is evi- 

 dent that a surface generated by the revolution about the axis 

 of x of a line connecting A and B might be of given superficial 

 extent, and at the same time enclose as small a solid as we 

 please; for instance, if the line were a portion of an ellipse of 

 great eccentricity, with its major axis parallel to, and extending 

 beyond, the straight line joining A and B. And it is equally 

 evident that the surface cannot enclose as large a solid as we 

 please. There must, therefore, be a particular surface which 

 encloses the limiting maximum. Not perceiving any reason for 

 conceding that the discovery of this maximum is out of the reach 



