94 
These are the equations of the sides of the inscribed lozenge in the 
directrix. 
The section of the surface by xz being itself one of the lines of 
curvature, this lozenge circumscribes it like the others. The points 
of contact of the sides of the lozenge with it may be found by 
eliminating z by their equations. The result is 
Baie 
Hence the points of contact are the umbilical points of the ellip- 
soid. 
The delineation of the projections of the lines of curvature of the 
ellipsoid on the plane of the greatest and least axes, is represented 
in Fig. 3. 
Prop. 
To determine the lines of curvature of the double surfaced 
hyperboloid. 
The equation of this curve may be derived from that of the el- 
lipse by changing the signs of a’ and 5°. It is therefore 
abs —Bb'ca—a'cy’ = abe 
The quantities in the last proposition expressed by e and f become 
in this case 
_ Va’ +e) 
~ a'(b’ +c’) 
B'(a*—b’) 
be+e¢ 
