101 
To fulfil the last equation it is necessary that a? and b? should have 
different signs. In this case the directrices are therefore two 
conjugate hyperbolas, the squares of the semiaxes of which are 
2 
1 
Ac= E. m 
B? i m-+ 1 
Tae ee te 
Since a’m=p” these hyperbolas are similar to the hyperbolic 
sections of the paraboloid perpendicular to the axis. 
With a’ and 8’ as semiaxes, let conjugate hyperbolas be described. 
Let ca= a’ (Fig 8.) and ch = 8’. The coordinates of any point p 
in either hyperbola being drawn, let an hyperbola be described with 
these as semiaxes, the transverse semiaxis being that which coincides 
with the second axis of the directrix. This will be the projection 
of one of the lines of curvature. In like manner points assumed in 
the other hyperbola will determine another system of hyperbolas, 
which are the projections of the other system of lines of curvature. 
As the points p approach the vertices a and b, the hyperbolas approach 
the axes aa’ and bb’ extending themselves nearer to the axes of # and 
y, until the actual coincidence of the points p with a and 5, 
when the hyperbolas actually become these lines. Which proves 
that the two sections of the surface by the planes wz and yz are lines 
of curvature. 
The general delineation of the projection of the lines of curva- 
ture of the hyperbolic ee upon a plane peependicula to the 
axis is represented in Fig. 
