Equations of the Second Order, 37 



which, from considerations analogous to those adduced in the 

 preceding problem, may certainly be pronounced to be possible. 

 First it is to be remarked that here, as in the last problem, 

 Apdx is an exact differential, while kdx is not, and consequently 

 that the equations Ajo = and A = 0, not being equivalent, 

 must both be taken into account. From the former we have 



Hence, by integration, 



b 2 being the arbitrary constant. Consequently 



d ,r- V-W* 



(4ay-(f-b*0 : 



This equation cannot be exactly integrated ; but MM. Delaunay 

 and Sturm have proved [Journal de Liouvillc, vol. vi. p. 315) 

 that the curve it represents maybe described by the focus of an 

 hyperbola which rolls on a straight line. Its form can therefore 

 be ascertained without difficulty. But on doing this it is found 

 that the curve is incapable of giving a solution of our problem, 

 whether or not the given points are on the axis. It has, however, 

 been shown by Mr. Todhunter (Hist, of Calc. of Var. p. 410), 

 for the case in which the problem is to find the greatest solid of 

 revolution of given superficies, the generating line of the surface 

 consisting of ordinates perpendicular to the axis at two given 

 points and of a curve connecting them, that the form of the 

 curve is given by the integral of the above equation, and that 

 the two ordinates arc equal and join on to the curve continuously. 

 But the general problem under consideration requires for its so- 

 lution an independent integration of A = 0. 



In order to effect such an integration, I shall proceed, just as 

 in Problem II., to deduce from A = the equation of the evolute. 

 Putting p for the radius of curvature, the equation A=0 gives 



Also 



Consequently 



^l+jo 2 J \a p) 



y-y- 



q Vi+P 



a v i 1 v(p— a ) 



y- 77-^ — y and w— y'= 4r^ -• 



* 2a— p' y J 2a— p 



^pyCa-l} 



