176 MR. ROBERT RAWSON ON SINGULAR 



* 



which is a differential equation having for its complete 

 primitive equation (10). 



Equation (n) is not, however, an exact differential, as 

 the multipliers which would make it so are 



1 ...&c. 



The first, second, &c. factors of (i i) are evidently satis- 

 fied by the relations 



w 1 = o, 

 w z = o, 

 &c. &c, 



(12) 



which are, therefore, singular solutions of (n) if they are 

 not contained in the primitive (io) by giving a constant 

 value to (c). 



By this assumption of the values of It,, R 2 . . . &c, it is 

 seen at once that the differences R x — B 2 , R 3 — R 4 . . . &c, 

 are made to vanish by the conditional equations (12) ; 

 hence the truth of theorem (A) . 



Since the equations (12) make the roots, with respect 

 to [p] in the quartic factors which compose (11) equal, 

 therefore the theorem (B) is also proved. 



The singular solutions here considered are of the enve- 

 lope species, as they may be obtained by eliminating (c) 



dii 

 between (10) the complete primitive and -j- = o. It will 



be seen that the integrating factors — :=> — ?=. &c. are 



made infinite by singular solutions, which is a well-known 

 property. 



5. In this paragraph there will be proved some of the 

 properties of singular solutions by means of the condition 

 of equal roots. 



