130 THE ROYAL SOCIETY OF CANADA 
(19) U = U3, v= V3, Us = Fue Vo — | Fake 
x = 22, y = Ax, 2 = dy, 
and subsequently drop the bars, the resulting equations are identical 
with (18). 
Remarks. 
1°. It is to be noted that since U’, V’ of equations (13) are distinct 
from zero, the functions U, V in the equations resulting from the 
transformation of parameters in (19) are of the third degree at least. 
2°. If in (18) or (13) after performing the transformation of 
parameters (19) we put 
U = Q2(u), V = Q'2(v), 
where Q, and Q’, are quadratic functions, the integral surface is a 
ruled quadric. 
3°. If in the same equations we put 
= Q2(u), = Q3(v), 
where Q, is a quadratic function and Q3 is a cubic function, the follow- 
ing invariants of the Normal os ses 
92) = aa — 1285 (<5 +202 +25 
= }? ou 
a none a (at i 
0? log a’ 
GO) QU—= eS ? 
Q Q Aaa 4a’b. 
But when these invariants vanish the integral surface is a Cayley cubic 
scroll.* 
Geometrical considerations. 
The parametric curves are asymptotic curves on the surface (18), 
since the characteristics of a Monge equation are asymptotic curves 
on an integral surface of the equation. Let (yz’), (zx’), (xy’), etc., be 
the Pliickerian coordinates of a line; then we shall find from equa- 
tions (18) that the asymptotic curves #=const., v=const. on the 
integral surfaces of the Monge equation (B) belong to the linear 
complexes 
si Kay) -246— EG 2)= 20 yy —y), 
an 
À (xy") + 2u(x — x") — (2—2') = — 2V"(y—y’) 
respectively. 
The axis of a complex being that diameter which is perpendicular 
to its polar planes, it follows that the axes of these complexes are 
given by the equations 
(21) Ax = —2U’, Ay = — Qu, 
and 
Ne = 2 Ay = = 20 
respectively. 

*Wilczynski, Trans. Am. Math. Soc., Vol. viii., p. 250; Vol. ix., p. 89. 
Sullivan, C. T., Trans. Am. Math. Soc., Vol. xv., pp. 175, 191. 
