[SULLIVAN] PARTIAL DIFFERENTIAL EQUATIONS 131 
The principal parameters of the complexes (20) are constant, 
and the axes (21) of these complexes are parallel to the z-axis. 
Inversely, if a surface be such that the tangents to its asymptotic 
curves belong to linear complexes of constant principal parameters 
and having their axes parallel to the z-axis (any line may be taken as 
the z-axis), then it must be an integral surface of the Monge equation 
(B). 
Let us consider the asymptotic curves #=const., and let the 
aig 1 
constant value of the principal parameters of the complexes be SR 
Since the axes of the complexes are parallel to the z-axis, the equation 
of the complexes must be of the form 
(22) Gy) aa MER Oy) eZ), 
where a, 8, y are functions of #. From the condition imposed on the 
principal parameters it follows that y (wu) = a . Hence the equation 
À 
of the complexes becomes 
(xy') = ax — 21) +87 — 9) + (2-2). 
Now the polar plane of a point P(x, y, z) on the surface proposed 
in the complex (22) must be an osculating plane to the asymptotic 
curve # = const. through this point. Hence the polar plane of P(x, y, 2) 
in (22) must be the tangent plane to the surface at P. On identifying 
these two planes, we find 
(23) he men UE) 
Oz 
SS Sa Takes meee 
If we consider a as an arbitrary function of 8, we have 
re = q 
G y) = 5 ( x +a ). 
ph 
which is the Monge equation considered. 
Let us now consider a surface .S whose equation is 
z2=f (x, y). 
If through the point (x, y, 0) a line be drawn parallel to the normal 
to S at P(x, y, z), the totality of these lines constitute a congruence G 
which we shall call the associated congruence of the surface S. And 
inversely, if the nornfal to a surface S at the point (x, y, z) be parallel 
to the line of a congruence G through the point (x, y, 0), we shall call 
S the associated surface of the congruence G. 
The congruence G associated with the surface S is generated by 
the line 
f+ po =x, 
24 
oe na gs Y; 
where (£, 7, ¢) are current coordinates and p = 
Hence 
Oz Oz 
ae cae. dy 
