of Differential Equations. 525 



which may be fairly said to be due to Lagrange, for the deduc- 

 tion of singular solutions from the integral expressions, applies, 

 in the case of partial differential equations in general. 



It will be remarked, too, that in the generalized form of 

 Clairaut's equation, the result of the elimination indicated for 

 the determination of the singular solution required is the same 



as the result of the elimination of -r, -r, -r, &c. between the 



ax ay az 



original equations and its derivees with respect to these quantities. 

 Examples. 



I. Let it be proposed to investigate the singular solution of 

 the partial differential equation 



du du du ^ ^ f ,/du\"' ^^{du\^ /du\'" „ "l 1 



The primary or general solution of this equation is, as we have 

 just seen, 



I 

 aa; + ^y + y2+&c. ={ a'"«'" + b""^'" + c"'j'" + &c. }m, 



whei-e «, ^, 7, &c. are arbitrary constants as many in number as 

 the variables x, y, z, &c. 



Differentiating this equation with respect to a, /3, 7, &c. seve- 

 rally, and denoting the expression within the brackets in the 

 right-hand member by U, we get 



Hence, by an obvious process of elimination, we obtain as the 

 desired singular solution, 



"» m m 



We shall find that it is only necessary to substitute, in this 

 result, particular values for m and limit the number of variables, 

 in order to obtain not merely the singular solutions of the 

 greater number of the differential equations, total or partial. 



