Prof. Cayley on Differential Equations and Umbilici. 447 



that is 



(2 m 2« m _ 1 ) A 2 + (2 m 2_ 3w + 1 )^ - (2m-2)A{A+(2m-l)v / D}=0; 



or 



and therefore 



A2+(2m-l)2/ 2 + (2m 2 -2m)A(^ + 2\/n) = (2m 2 -m)(A 2 +y 2 ). 



Hence the term in j | is 



= (2m 2 -m)(A 2 +2/ 2 )(A + ^); 



or, what is the same thing, it is =(4m 2 — 2m)A<s/ D{A + yp). 

 Hence, restoring for A 2 + (2m— l)?/ 2 its value 2mB, we find 



«=* ^ (A + jjp), 



or 



But writing U lf U 2 to denote the values corresponding to 

 +\/D, — a/d respectively, we have 



■"1 



TT/ (2m-l)U 2/ ,_ 



Or, since 



■U'=P'+QV"D + 37^ = 27^(2Q'I=> +Qn'+2P'v / D, 



and thence 



U , itJV=-^j{(2Qfn +Qn') 2 -4P' 2 D}, 



and moreover 



U 1 U 2 =P 2 -Q 2 D =A 1 A 2 (B 1 B 2 )r 1 

 where 



B 1 B 2 =(m^ 2 + 2 / 2 ) 2 - □ =2/ 2 {2/ 2 +(2m-l)^. 



and thence 



