Equations by Factors and Differentiation. 397 



that the connecting curve joins on to the ordinates continuously. 

 The discovery of this discontinuous solution, which differs in kind 

 from that discussed above, is due to Mr. Todhunter. 



We have not, however, as yet arrived at a solution which gives 

 a single continuous curve joining the given points — although, 

 from considerations exactly like those adduced in the case of the 

 preceding problem, it is certain that such a solution exists. 

 The difficulty, as in that case, is overcome by employing the 

 proposed new method of integration, which in the present in- 

 stance consists in first deducing from the equation (8) the dif- 

 ferential equation of the first order of any curve parallel to the 

 curve traced by the focus of the rolling hyperbola, and then 

 showing that the equation (77) resulting from the elimination of 

 the constants from that differential equation is verified if F=0 



and -t- = 0, the function F being y -. -\ ™ . . The 



doc ' s/\+f (1+p*)* 



parallel curve, as 1 have already argued, is under these circum- 

 stances an integral of F = ; and since its equation contains one 

 more arbitrary quantity than the equation of the curve itself, it 

 is capable of satisfying the given conditions of the problem. I 

 have ascertained, for instance, that if the given surface be of 

 small amount for a considerable distance between the given 

 points on the axis, the form of the curve is like the arc of a 

 bow ; but if the amount of surface be large for a small interval 

 between the points, the curve approaches the circular form. 

 Under the supposed circumstances such forms might be ante- 

 cedently expected to be given by the results of the analytical 

 calculation. 



The foregoing solutions of the two problems do not essentially 

 differ from those I proposed in an article contained in the Phi- 

 losophical Magazine for July 1871. But the latter solutions 

 were effected by actually finding the equations of the evolutes 

 corresponding to the two series of involutes ; whereas the present 

 research has shown that this process was not necessary, and that 

 the new integration is virtually an extension of the recognized 

 methods of integrating by factors and by differentiation. The 

 principle of the application of this new method in these two in- 

 stances may be briefly stated as follows. A differential equation 

 F=0 of the second order may have an integral containing three 

 arbitrary constants, but only on the condition that the differen- 

 tial equation of the third order obtained by eliminating these 

 constants from the integral is verified by the two equations F = 

 dF 



and — =0, and not by either singly. The reason is, that as 



. dF 

 the equation -=- = is a necessary consequence of F = 0, the 



