Then 



Equations of the First Order, 127 



Take the well-known correlated pair 



u=p, v = xp — y, and w — x. 



px + ay = (l+a)uf l u — afu, 



(xp-vY = (far 



px + ay ( 1 + a)uf l u — afu 



Therefore 



is solved by 



(xp-y) m . 

 px-\-ay T Jr 



v=fu, 

 where fu is to be derived from the equation 



(\+a)ufu-afu Y ' 

 and this is easily found by making 



a 

 fu=^Ul + a^U, 



which gives 



l+2ffl 



(1 + a)ufu —afu = (1 + a)u 1+a %'w, 

 and 



l+2g m(a— 2)— 1 



(\ -{■ a)u l+a ^U _ 1 %'^ 1 W a+1 . 



^i —: ii^? 01 (x u ) m "~ 1 + « ^M ' 



which is explicitly integrable. 

 If we take 



w=p# a+1 , v = xp-\-ay } w=x~ a , 

 we find 



xp + % 5/w + (a—b)uf 'u' 

 so that the more general form 



(xp + ay) m = (ajp -j- by)i]r(px a+l ) 



is integrable. 



Some of the examples which I have given above may be found 

 in an elaborate memoir by M. Malmsten, contained in the 

 seventh volume of the current series of Liouville's Journal. In 



