226 Mr Greenhill, Note on Prof. Cayleys paper on [March 20, 



For other values of u + v besides §K, the integral of the 

 differential relation 



dx chi 



+ /T~M = ° 



(1-^)1 (l-/)3 



leads to a cubic-cubic relation between x and y, which can be 

 obtained by the elimination of t between the equations 



1 + x _ 4/n sn u dn u 



1 — x *" v (1 + cnuy 



l-y v (1 + cnv) 3 ' 

 where u = a — t, v = a + t, and 2a is the constant value of u + v. 



More generally, the integral of the differential relation 



dx dy n 



+ — — — z—z — — — a = 



(A + SBx + 3Cx 2 + Da?f (A + 3By + 3 Gif + Diffi 



has been obtained by Captain MacMahon, R.A., in a rational form 

 as a cubic-cubic relation between x and y ; instead of in the irra- 

 tional form as left by Allegret (Comptes Rendus, t. 66) ; his result 

 assumes the symmetrical form 



{A + B (x + y + z) + C (yz + zx + xy) + Dxyz} 3 = 

 (A + ZBx+ZCx^+Dx^A+Wy+SCf+Dy^iA+BBz+SCzt+Dz 3 ), 



where z is the arbitrary constant. 



The general integral of 



*" +A1-01 



will therefore be 



(l-xyzf=(l-of)(l-y 3 )(l-z 3 ); 

 or y 3 z 5 + .2 V + x 3 y 3 - 3x 2 y 2 z 2 - x 3 - y 3 - z 3 + %xyz = 0, 



where z is the arbitrary constant ; and 

 when z = 00 , x 3 + y 3 — 1 = ; 



when 2 = 1, #^ — 1 = 0; 



when z = 0, -= + -, - 1 = 0. 



x 3 y 3 



