87 
scribed in the same manner.. ‘The ellipse and hyperbola thus de- 
scribed will be the projections of lines of curvature of the given 
ellipsoid, and in the same manner the projections any number of such 
lines of curvature may be found. The ellipse and hyperbola with 
the common axes 2a’and 2p’ may be denominated the directrices of 
the lines of curvature. 
The greatest axis of the ellipsoid being 2a, and the least 2c, it fol- 
lows that a?—b? < a’—c?; and therefore a’ < a. Hence the com- 
mon vertices of the directrices fall within the ellipsoid. 
Let sa’s’ (Fig 1) be the section of the ellipsoid by the plane of 
ay, and ca=a, cp =b. And upon cA assume ca = a’ and upon 
cs produced, ch = 3’. Let the ellipse aba‘b’ be described and the 
hyperbola, pad, v‘éd’, with the same axes. Any point p being assumed 
in the ellipse aba’b’ and its coordinates pm, pm’ being drawn, let 
an hyperbola be described with the lines cm as semitransverse axis 
and cm’ as semiconjugate axis. The branches of this hyperbola 
will be the projections of two. lines of curvature Let tangents be 
drawn through the vertices a,a’ of the elliptic section, and any 
point p on the hyperbolic directrix, between these tangents Té and 
rt’ be assumed. The coordinates pm, pm’ of this point being drawn, 
let an ellipse be described with cm and cm’ as axes. This ellipse 
will be the projection of another of the lines of curvature upon the 
plane zy. 
In like manner other points being assumed upon the ellipse aba’b’ 
or hyperbola tar, any number of lines of curvature may be deter- 
mined. As the point assumed upon the ellipse approaches the ver- 
tex b, the coordinate cm diminishes and cm’ increases, therefore the 
transverse axis of the hyperbola continually diminishes, and the 
conjugate axis continually approaches to equality with bb’. When 
the vertex 6 itself is the assumed point, the transverse axis va- 
