192 MR. ROBERT RAWSON ON SINGULAR 



From these equations (v), the only unknown element in 

 the primitive c + v + V w = o, when */w is given, can be 

 easily found by the usual process, as given by Boole (see 

 Diff. Eqs. pp. 48, 49, 2nd ed.). If, however, the solution 

 w = o has unfortunately dropped a factor, it will be a 

 serious matter, in many cases, to pick it up, as wz z = o 

 must be used in (61) and (62) instead of w = o, where z is 

 an unknown quantity to be determined. 



It is therefore obvious from the primitive that the solu- 

 tion w = o cannot be particular if (v) contains x or y, 

 except in the case of (v) being a function of (w). 



If the dexter side of (61) and (62) are functions of x, y 3 

 the solution iv = o is singular and of the envelope species. 



Example : — 



xp z — 2yp + 4.x=o (from Glaisher), . . (63) 



Compare this example with (58), (61), (62) ; then 



2?/ 2 /—„ r 



r-\-s= -_, rs=4-, s—r=-\/y z —4.x. 



30 W 



The condition of equal roots of (63) is y= + 2x, which 

 is a solution of (63) ; then 



w =y z -4.x\ 



dw 



dw 



Substitute these values in (61) and (62), then 



dv dv ., r 



-r- =0, —- = 1, therefore v=y. 



dx ' dy ' J 



