89 
aa’. Hence it appears that the concayities of all these curves must 
be presented towards the points a, a’. The major limit of the con- 
jugate axes of the hyperbolas is ch, and that of the similar axes of 
the ellipses is cz. The points a, a’ are the projections of four points 
of the surface towards which the concavities of all the lines of cur- 
vature are turned. These points are upon the section of the surface 
by the plane zz. This section being itself a line of curvature all 
other lines of curvature which meet it intersect it at right 
angles. ‘The projections of those which intersect it between the ver- 
tices of the axis 2c and the points whose’ projections are a, a’y are 
hyperbolas. The projections of those which intersect it beyond 
these points are ellipses. ‘Thus the lines (of curvature perpendicular 
to the section by the plane wz change their species as they pass these 
points, presenting their concavities on both sides towards these 
points. These are called wmbilical points of the surface. 
The umbilical points of any surface may be determined: ‘by con- 
sidering that they are those points at which the two lines of curva- 
ture interchange their species, and at which they therefore coincide. 
To find these points it will be only necessary to investigate under 
what circumstances the roots of the equation which determines 
d: 
the value of <> for the lines of curvature, are equal, In the present 
case this equation is 
-- e=0 
dy? y wf t ay 
isles ato da 
The condition of the equality of the roots is 
y @ Sf = 2 
(bred eZ) 4 te 20 oy nent +f) 4 Aea'y' = 0 
