98 
Let ca = a’, ch = 8’, (Fig. 6) and let the directrices be described 
with these lines as axes. Any point p in the elliptic directrix being 
assumed, and its coordinates drawn, with these, cm and cm’, let an 
hyperbola be described; this will be the projection of one of the 
lines of curvature. In like manner a point p in the hyperbolic di- 
rectrix will determine an ellipse, which will also be a projection of a 
line of curvature. As the point p approaches 6 the transverse axis 
of the hyperbola diminishes and vanishes when p coincides with b. 
At this point the projection of the line of curvature is the axis bb’, 
and therefore the section of the surface by the plane yz is a line of 
curvature. When p coincides with a, or a’ the hyperbola closing 
its branches, becomes a straight line and coincides with the axis aa’. 
So that the section of the surface by the plane zz is a line of curvature. 
It may be observed generally, that those parts only of the ellipses 
hyperbolas and right lines thus determined, are actual projections of 
the lines of curvature, which fall outside the elliptical section of the 
surface by the plane ay; for the value of = for every point within 
this elliptical section is impossible. Hence it follows that although 
at the points a, a’ the projections of the lines of curvature change 
their species, and although all their concavities are turned towards 
this point, there is no corresponding umbilical point in the surface. 
The delineation of the projections of the lines of curvature of the 
single-surfaced hyperboloid upon the plane of its real axes is repre- 
sented in Fig. 7. 
Prop. 
To determine the lines of curvature of the elliptic Paraboloid. 
The equation of this surface is 
y+mx* = pz 
