132 THE ROYAL SOCIETY OF CANADA 
The projection on the plane z=0 of the net of curves on S which 
corresponds to the developables of G is found by eliminating (£, 7, €) 
between (25) and the equations 
(25’) E + (p+ rdx + sdy) £ = x + dx, 
n+ (g + sdx + dy) £ = y + dy. 
Thus this net of curves is given by the equation 
(26) ere: 
The planes tangent to the two developables of G that pass through 
the line (25) are also tangent to the focal surface of G. The directions 
of the traces of these planes on the (xy) plane are given by equation 
(26); their equations are then 
[E—@ = 20) —Aln— (y¥ —9h)] = 9, 
where ) is a solution of the equation 
Soe 
Ss 
Since the roots of the latter equation are real and distinct, the focal 
surface of G has two distinct sheets. 
If the ray (25) be tangent to the focal surface at P and Q and M 
be the middle point of PQ, the locus of M is called the middle surface 
of the congruence (Eisenhart, Differential Geometry, p. 399). 
Let us now consider the congruence G whose middle surface is 
the simplest type, viz., a plane. We shall find that the integral sur- 
faces of the Normal System (A) or what is the same thing, as we have 
seen, the integral surfaces of the Monge equation (B) are associated 
surfaces of the congruence contemplated. 
In short we find from equations (25) and (25’) that 
¢ (rdx + sdy) = dx, 
£ (sdx + tdy) = dy, 
and therefore 
(27) c+ = on 
st — pl 


: 2 t 
The altitude of the point M above the (xy) plane is —3 (Gane 
If the middle surface of G be the plane (g=0), the associated sur- 
face S must be an integral of the Laplace equation 
072 d?g 
(28) he vee JU ES 
Now (28) is precisely the condition that the net of curves obtained 
by projecting the asymptotic curves of S on the (xy) plane may be 
orthogonal; because the directions of the curves of the net through 
the point (x, y) are given by the equation 
rdx? + 2sdxdy + tdy? = 0; 
and these are orthogonal if and only if (x, y) be an integral of equa- 
tion (28). 
— ] 
If, however, the middle surface of G be the plane (: = ee 
