100 
trices are the projections of two umbilical points upon the plane 
zy. These points are therefore in the parabolic section of the 
surface by the plane az. Since the values of « for these points are 
m—l1 
+ — . Jnl andl y=0, -.8= t sj” «Hence to find the umbili- 
m 
cal points let a section of the paraboloid be made by a plane per- 
1 ; 
from the vertex ; this 
t ; é m— 
pendicular to the axis at the distance pe 
section will be an ellipse, and the umbilical points will be at the ex- 
tremities of its lesser axis. 
The delineation of the projection of the lines of curvature of the 
elliptic poraboloid upon a plane perpendicular to its axis is similar 
to that of the ellipse, and is represented in Fig. 2. 
Prop. 
To determine the lines of Curvature of the Hyperbolic Paraboloid. 
The equation of this surface is 
yr—ma2* = —ps 
In this case, as before, m may be considered > I and p > 0. 
The lines of curvature are to be determined by the equations 
ay? + b2n? = a? b? 
p” : m+1 
4 m 
ma? +b? =.— 
