88 
nishes, and the conjugate axis. becomes equal to bb’, with which both 
branches of the hyperbola coincide. Hence the axis bb’ is the pro- 
jection of one of the lines of curvature, which must therefore be the 
section of the surface by the plane yz. 
As the assumed point approaches the vertex a, cm increases and 
cm’ diminishes, therefore the transverse axis of the corresponding 
hyperbola continually increases, and the conjugate axis diminishes. 
When. the.assumed point, is the vertex a itself, the conjugate axis 
vanishes, the transverse axis is aa’, and the branches of the hyper- 
bola closing become coincident with the right lines aa and.a’a’. 
As the point p assumed upon the hyperbolic directrix approaches 
the vertex a, the conjugate axis of the corresponding ellipse dimi- 
nishes as well as the transverse axis. When P coincides with the 
vertex a, the conjugate axis of the ellipse vanishes, and the transverse 
axis becomes equal to aa’ with which the ellipse itself coincides. 
This axis of the surface being therefore the projection of one of the 
lines of curvature upon the plane vy, this line must be the section 
of the surface by the plane az. 
As the point p approaches the tangent Tt, the axes of the corres- 
ponding ellipse both increase. When p coincides with the tangent 
at T the axes become equal to those of the ellipse aBa’p’, with which 
the ellipse then. coincides... This ellipse therefore, the section of the 
surface by the plane zy, is one of the lines of curvature. 
When the point p passes the tangent, the corresponding ellipse in+ 
cludes the given surface within it, and therefore cannot be the pro- 
jection of any line in the’surface. 
It is plain that the transverse axes of all the hyperbolas which are 
projections of lines of curvature are less than aa’, the common axis 
of the directrices ; and all the transverse axes of the ellipses which 
are projections of lines of curvature, are greater than the same line 
