of Differential Equations. 531 



The general solution is 



aa {ux + ^1/ + 7^ + &c.) +>y32 + cV + &c. = 0, 



where, as before, a, /S, 7, &c. are arbitrary constants, and the 

 singular solution desired is readily determined to be 

 «2 g'i yji 4^ 



0* c^ d^ a 



When the number of variables x, y, z, &e. is reduced to three, it 

 appears then that the singular solution of the equation 



du( du du du\ .o/^mX^ „{du\'^ ^ 



'T.VTx-^ydy+'Tz)+^\dy) +^ b; =° 



is 



y^ z^_4x 

 b^^ c^~ a' 



a result which is at once verified by the known properties of the 

 paraboloid. 



2. Thus far we have been employed in the derivation of the 

 singular solutions of partial differential equations from their 

 general or primary integrals, the method being suggested by an 

 easy modification and subsequent generalization of Clairaut's 

 theorem, and the application of the method being illustrated by 

 examples sufiiciently numerous. I now proceed to consider a 

 general method which has been proposed for determining the 

 singular solutions of differential equations, both total and par- 

 tial, from the equations themselves, without the knowledge of the 

 geneial or primary solution. 



The following theory for differential equations of the first 

 order has been attributed to Laplace: — "Let U = be a differ- 

 ential equation of the first order between x and y, cleared of 



dv 

 radicals and fractions ; then if we represent -j- by p, the rela- 

 tions between x and y found by eliminating p between 



U=0, f =0, 



are singular solutions of U = 0, jjrovided they satisfy that equa- 

 tion and do not at the same time make 



. . . dx 

 we might also deduce the singular solutions by eliminating -j- 



between TT — n '^^ — n 



