95 
Since a? > be, f is essentially negative. . The lines of curvature pro- 
jected upon the plane zy are represented by the equation 
a’y? + B’ a" = a”h” 
where a” and b” are connected by the relation 
ea?—b” = f 
The semiaxes of the directrices are 
ieee /a—B 
Vate 
ae 
B= b va—h 
J b+ ce 
The transverse axis of the hyperbolic directrix coincides in this case 
with the axis 2b, The equation of the elliptic directrix being 
12, 2 
A’y’ + Ba = a?” 
and that of the hyperbolic 
A’y’—B'a? = a°B” 
With these common axes let an ellipse and hyperbola be described 
upon the plane wy. As before with the coordinates of the points of 
the hyperbola as semiaxes ellipses being described, and with those 
of the points of the ellipse as semiaxes hyperbolas being described, 
these ellipses and hyperbolas are the projections of the lines of cur- 
vature upon the plane ay. 
Let ca, ch, (Fig. 4) be the semiaxes of the directrices, and let any 
