of Differential Equations. 523 



this theorem as a practical truth admitting of large and useful 

 application, and the still higher difficulty of generalizing it with 

 a view to the integration of corresponding classes of partial dif- 

 ferential equations in any number of independent variables, and 

 the simultaneous determination of their singular solutions, seem 

 to have arisen in a great measure from the unsymmetrical form 

 in which this theorem happens to have been presented. 



Stated in a symmetrical manner, the differential equation with 

 which Clairaut was concerned assumes the shape 



du du T^ ( du du\ „ 



where F, is any homogeneous function of the first order in the 

 quantities included within the brackets. 



At a glance we perceive that the general solution, m=0, of 

 any differential equation of this type is 



«^ + /3y-F,(«,/3) = 0, 



where a and /3 are any arbitrary constants, and that the singular 

 solution is due to the elimination of a, /S between this general 

 solution and the system of equations 



But, more generally, we see that if we were given any partial 

 differential equation of the type 



du du du „ /du du du\ „ 



its general solution, u = 0, is 



where «, /3, 7 are any arbitrary constants, and that the singular 

 solution is due to the elimination of «, /3, y between this general 

 solution and the system of equations 



du 



z r^ = 0. 



dj 



It will be remarked, that in the previous case, although there 

 are in appearance two arbitrary constants, there is in reality but 

 2 M2 



