SOLUTIONS OF DIFFERENTIAL EQUATIONS. 175 



makes the multiplier, or the integrating factor, infinite; 

 hence from (5) the singular solutions of (1) are 



(R I -R 2 ) i ...(R I -E, l ) i x(R 2 -E j ) i ...(R z -U n ) 4 



x(R J -R 4 )\..(R ? -R n ) i &c. = o. ... (6) 



Of course each of the factors must be equated to zero, 

 therefore the theorem (A) is obvious. 



From (3), if R z = R a then will r 1 = r % &c, therefore the 

 truth of theorem B is manifest. 



4. In consequence of the importance of the principle of 

 the condition of equal roots in the determination of sin- 

 gular solutions of differential equations, another proof of 

 it may not be deemed unnecessary. 



In the primitive (4) the functions R„ R z . . . . R„ can 

 be made to assume various forms; put, therefore, 



B»i=Vi+ \/iv I} U 1 =v I — \Sw l} ... (7) 



R 3 =v z + \/w z , R 4 =v z — \/w z , ... (8) 

 R 5 =S7 3 4- sjw v Tt 6 = Vi — sjw v ... (9) 



&c, &c, 



where v t , iv T ; v %J iv z ; v v w v &c. are functions of x, y. 

 The primitive (4) becomes by substitution 



\(c + v I ) z —w 1 }{(c + v 2 ) z —w z \{(c + v i ) z —iv z }...&c. = o, . (10) 



where (c) in each quadratic factor has a two- fold value. 



Differentiate (10) with respect to x, eliminate (c), and 

 reject the common factors ; then 



I 



— ax 



dw t 

 dx 



1 — dVj 

 ±2 s/w 1 -j J - 



dw l 



, — dv, dwr\ 



+ 2 Vw,-,- r 1 



dy f —" y ' ™ l dx dx [_ J dy — " y "" z dx dx I 



dx + i--*rH 



dx ,— x dv x dw x 



- W dy dy J 



. . . . &c. = o, . . . (11) 



