90 
Since e is a quantity essentially positive, this condition can only 
be fulfilled by 
ay=0 
y —ex?+f=0 
If y = O and therefore z — Vi these are the coordinates of 
the umbilical points already determined. If x=0..y =./—f. 
Since f is positive, this value of y is imaginary. Hence there are 
but the four umbilical points already determined in the ellipsoid. 
The delineation of the projection of the lines of curvature of the 
ellipsoid upon the plane of the greatest and mean axis is represented 
in Fig. 2. 
To determine the projections of the lines of curvature upon the 
plane zz, let y be eliminated by the equations 
a®h’z: + acy? + b'c’x” = a*b'c® 
a*y’+ b? x? = a’ 2 
This elimination being effected, and b” eliminated by the equation 
ea*—b* = f, 
The result is 
c’'(a’—a")fa* +. a*b'a’z’ = ea’e’a* (a’°—a* ) 
The process of elimination will be facilitated in this case by con- 
sidering that 
b+ f= ae. 
The equation thus obtained is that of the projection of the lines 
