174 



MR. ROBERT RAWSON ON SINGULAR 



where R„ R 2 . . . R n are functions of as } y, which must 

 satisfy the following partial differential equations, viz. 



dR, dR T ^ 



dx 



1 dy 

 dR, 



dx 2 dy 



dR n _ dR r 

 dx 



> 



(3) 



n dy 



j 



And c 1} c z . . . . c n are arbitrary independent constants, 

 which, however, may be made to satisfy (n — i ) -fold 

 relations. 



3. An interesting case of the above is where each of 

 the values of c t , c z . . . . c n is represented by the arbitrary 

 constant (c) ; then (2) may be written 



(c + ll 1 ){c+'R 1 ) .... (c + R n )=o, 



(4) 



where R I} R z . . . . R n are the roots of the primitive (4) 

 with respect to the arbitrary constant (c). 



Equation (1) in its present form is not generally an 

 exact differential equation of the primitive (4) ; the multi- 

 plier, however, which will make it exact is readily obtained 

 by differentiating (4) with respect to x, and will be as 

 follows : — 



-• (5) 



(^-RJ 2 . . . (R x -R„)* x (R z -R,) 2 • • • (J^-ftJ 2 



x (R 3 -R 4 )\ . . (R 3 -R„) a . . . x (R 4 -R 5 ) 2 . . . (R 4 -R„) 2 



Now it is proved by writers on the subject of singular 

 solutions of differential equations that a singular solution 



