99 
in which m is positive and > 1. If p > 0, the surface lies on the positive 
side of the the origin, and if p < 0 on the negative side. As this 
is a mere variety in the position of the surface, and none in its species 
or properties, it is unnecessary to investigate more than one of the two 
cases. Let p be therefore assumed positive, and the surface con- 
sequently on the positive side of the origin. 
The equations which determine the lines of curvature being 
a”y? + ba? = ah” 
ea’?—b” =f 
in which 
e=m 
ay P m—l1 
I a’ m 
The quantities e and f are essentially positive, and the squares of 
the semiaxes of the directrices are 
A 
4 me 
a_p,m—l 
23 4 m 
Hence a’m = B” and therefore the elliptic directrix is similar to the 
sections of the surface by the planes perpendicular to the axis, for 
m is the ratio of the squares of the axes of each of these sections. 
Let ca = a’, ch = B’ (Fig. 1.) and with these semiaxes let an el- 
lipse and hyperbola be described. With the coordinates of each 
point of the ellipse as semiaxes let hyberbolas be described and with 
those of each point of the hyperbola let ellipses be described, these 
curves will be the projections of the lines of curvature as in the 
case of the ellipsoid. The extremities of the axis aa’ of the direc- 
VOL, XIV. R 
