170 MR. ROBERT RAWSON ON SINGULAR 



XVII. On Singular Solutions of Differential Equations. 

 By Robert Rawson, F.R.A.S., Hon. Memb. Man- 

 chester Literary and Philosophical Society, Assoc. 

 I.N. A., Memb. of the London Math. Society. 



Read November 14, 1882. 



i. Singular solutions of differential equations have re- 

 ceived great attention by most mathematicians since Dr. 

 Brook Taylor announced the first example of them in 

 1715. 



This is not to be wondered at, as it is important to 

 know that differential equations, even of the first order 

 and of the nth. degree, may have other solutions than those 

 commonly called their complete primitives. 



In the application of the higher mathematics to physical 

 problems of practical importance there should arise a 

 relation between two variables x and y, with an arbitrary 

 constant expressed by the differential equation 



\dx) + 2 ( V^+2/+ V^y) J + 4 s/x z -if 



+ 2( s/'x-y — sfx + y) — 1=0, . . (a) 

 whose complete primitive is 



c rj \-{ \/x + y + s/x—y)c + \/x l —y z 



+ x( \'x— y— s/oc + y)— x z = o. . . (b) 



It must be important to know that (a) has two solu- 

 tions, viz. x + y = o, and x — y = o } besides the complete 

 primitive (b) , neither of which is a particular case of (b) . 



In the determination of singular solutions it has been 



