442 [ Prof. Cayley on Differential Equations and Umbilici, 

 we have 



§El=f. ov (p-v)(vp + \)=0, 

 and we may write 



Assuming also 



V 

 y — ux, or u— -, 



y ■ x' 



the relation between u and w is 

 b + cu —2v 



or, as this may be written, 



o + cw 



giving 



r= -(/+^)-v / (6 + ^)* + (/+^ 

 b-\-cu 



where for convenience the radical has been taken with a negative 

 sign. We have moreover 



fl(g«-l) + 2/t? 



* se c (v*-l)+2gv 



The equation p— v=0, substituting for y its value ux } then 

 becomes 



#-7- + u— v~0; 

 ax 



or, as this may be written, 



dx du 



x u—v 

 or, what is the same thing, 



dx dv—du dv _ 



x v — u v — u 

 But 



b{v*-l)+2fv _ V 



where 



Y=v[c{v' 2 -l)+2gv}+b' 2 (y-l)+2fv 



= (b + cv){v*-l)+2(f+c/v)v, 

 and the differential equation takes thus the form 



dx dv—du \_c(v 2 — 1) + 2gv\ dv _ n _ 

 x v—u V 



