446 Prof. Cayley on Differential Equations and Umbilici, 

 where it will be remembered that 



then we have 



2mB = A 2 + (2m-l)2/ 2 . 



The integral equation may be written 



^=P + Q v / P=U = AB m - 1 . 



And we have 



U' A' 1X B' e 



tr = x+ (m-l) s -= 7W , 



if 



Or, since 



©=Vd{x+(™-i)|j" 



and 



this is 



A\/ D = m\/ □ -{-m^x + yp^mA + yp, 

 B'\/"D = {2mx + 2yp)\/~n + D -f x(m?x + yp) 



=2m?x 2 +y 2 +xyp + (2mx+2yp)\/~n 



z=A 2 +pyx + 2\/n, 



1 2m 



B^A^^m-l)*/ 2 ' 



^mL+yp M2m z_ 2m) A*+py(x + 2^) 

 A v ; A 2 +(2m-% 2 



+ (2m 2 -2ro)[A 3 + A^(#+2\/a)]}, 

 where the expression in -j J- is 

 = (2m 2 -m)A(A 2 -j-2/ 2 ) 

 +2^{A 2 + (2m~% 2 + (2m 2 -2m)A(a?+2v / "a)} • 



Here the coefficient of yj9 is = (2m 2 — m)(A 2 + z/ 2 ) ; in fact we 

 have identically 



A 2 + 2/ 2 -2A v / D=0, 

 and thence 



(2m 2 -3m + l)(A 2 + 2 / 2 )-2(2m-l)(m-l)Av / D=0, 



