178 MR. ROBERT RAWSON ON SINGULAR 



Differentiate each factor in equation (n), and reject the 

 common factors ; there results for the first factor by drop- 

 ping the affixes, 



(dw dv\ i ,- 



— \ dy dx) k/w ~ 



- d z v d z w 



Us) 



dp ~ V dy dx) sj w ~ dxdy dxdy 



pdy , — dv dw 



1 J + 2 \/w-. r- 



~~ ax dy 



(dw dv\ i ,— d z v d z iv 



~\ly dy)~^7o~ fl S/W dy z ~~~aY 

 ,— dv dw 

 - dy dy 



For a singular solution w — o, then (15) becomes 



| =infinit y ( i6 ) 



if 



±( 



dv div dv dw\ 1 

 dx dy dy dx ) dw 

 dy 



is a finite quantity, and v a function of x, y. When, 

 however, the above quantity is zero, then -~- = - } an inde- 

 terminate quantity, and by Boole's Diff. Eqs. p. 24, 2nd 

 ed., v will be a function of iv, and thereby it is probable 

 that the solution will be a particular integral. 



6. As the differential equation of the first order and of 

 the ftth degree is composed entirely of the simple factors 

 of a differential equation of the first order and degree, it 

 will be necessary to consider the latter form only. 



Let 



fx + r = ° ( J 7) 



be a differential equation where r is a function of x, y. 



