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point p be assumed upon the ellipse ; and with its coordinates cm and 
cm’ as semiaxes let an hyperbola be described, and a point p being 
assumed upon the hyperbolic directrix let an ellipse be described 
with the coordinates cm and cm. ‘The ellipse and hyperbola thus 
described are the projections of lines of curvature, and in the 
same manner any number of such projections may be drawn. 
As the point p recedes from the vertex of the hyperbolic direc- 
trix, the ellipses continually increase, and they diminish indefinitely 
as it approaches that point. When p coincides with b, one axis of 
the ellipse becomes equal to bh’, and the other vanishes, and the 
ellipse coincides with the right line bb’. This line being therefore 
the projection of one of the lines of curvature, the section of the 
surface by the plane yz must be that line. 
As the point p passes through 6, and proceeds in the elliptic arc 
bp, one axis continually diminishes, and the other increases. 'The 
projections also change their species and become hyperbolas. The 
transverse axs of the hyperbola continually diminishes as the point 
approaches a, and vanishes when p coincides with a. ‘The hyper- 
bola then extends itself into a straight line, and coincides with the 
axis aa’. This being the projection of the section of the surface by 
the plane ez, that line is one of the lines of curvature. 
The points b, b’ correspond to four umbilical points upon the hy- 
perboloid. As in the ellipse the lines of curvature change their spe- 
cies in passing through these points, and their concavilies are every 
where turned towards them. 
The delineation of the projections of the lines of curvature of 
the doub e surfaced hyperboloid upon the plane of its imaginary 
axes is represented in Fig. 5. 
