SOLUTIONS OF DIFFERENTIAL EQUATIONS. 189 



whose complete primitive is 



c + F(a?,w;)=o, (50 



dw 



: 



yjr [x } w) 



= or oo . . . (52) 



makes w = o a singular or a particular solution. 

 Differentiate (51) with respect to x, then 



, -j- E (x. w) 



aw _ ax v _ ' 



ax a ._. , . 



Hence 



~j-~E(x,w) 



^{X,w)=:--j ; 



from which there results 



i 



dw I dw ' , , „ 



■tfu>. . . (53) 





cfo? 



The integral on the dexter side, as given by Boole, is 

 equal to H .~F(x, w), " where (H) is some value inter- 

 mediate between the greatest and least values of 1 upon 



-7-F(V, w) within the limits of integration/'' See supp. 

 ax 



vol. p. 29. 



This integral, however, is regarded as unsatisfactory, 



therefore the following is proposed in its place, viz. 



i 



■> • ■ ■ (54) 



f( X , W ) Jp^^) 



