396 Prof. C hall is on Integrating Differential 



line of the surface terminated at two given points not on the 

 axis ; but for the sake of simplicity I shall take the problem as 

 above enunciated. 



The history of the various attempts that have been made to 

 solve this problem is very instructive relatively to the principles 

 of the Calculus of Variations, especially as regards the distinc- 

 tion between continuous and discontinuous solutions. With 

 respect to the principle of discontinuity, it may first be remarked 

 that the integral of the equation designated as M = in Mr. 

 Todhunter's work (the same that I have usually called A = 0) 

 may either give one line, or two or more lines. But in such 

 cases there will be, according to Algebraic Geometry, a factor 

 corresponding to each line, and the answer to the question may 

 be given by parts of lines terminating at points where two lines 

 cross each at finite angles of inclination. This is the case in the 

 solution I gave of the Problem of the Brachistochronous course 

 of a Ship, which is considered by Mr. Todhunter (Chap. I., 

 art. 12) to be the first instance of a solution exhibiting this kind 

 of discontinuity. The problem before us presents another in- 

 stance of the same kind, inasmuch as we know beforehand that 

 the spherical form is that for which the volume for a given sur- 

 face is a maximum, and that consequently one solution of the 

 problem should be given by a straight line coincident with the 

 axis of revolution and a semicircle terminating at both ends in 

 that line. It is found in fact that this solution is deducible 

 from the equation (8) when h 2 is supposed to vanish, in which 

 case «/ = 0, which is the equation of the straight line, and 



y= , which is the differential equation of the circle. 



Vl+p 2 

 Moreover the differential equation ^>M = 0, which requires the 

 factory for effecting the integration, is satisfied by y = 0, the 

 equation of the straight line, which causes p to vanish, and by 

 ,2? 2 + 2/ 2 = 4a 2 , the equation of the circle, which causes M to vanish. 

 This discontinuous solution was given by the Astronomer Royal 

 in the Number of the Philosophical Magazine for July 1861. 



Again, as has been already stated, the integral of the equation 

 (S) gives the curve which is described by the focus of an hyper- 

 bola rolling on a straight line. But, contrary to what might 

 have been expected, this curve by no means gives the solution 

 of the problem. It is wholly inapplicable if the positions of the 

 extremities of the curve, whether on or off the axis, be given ; 

 but if the abscissae only of the extreme ordinates be given and 

 the surface generated by the revolution of these ordmates and 

 the curve joining their extremities be also given, the solution 

 shows that the form of this curve is that traced by the focus of 

 a rolling hyperbola, that the extreme ordinates are equal, and 



