84 
face by the plane zy itself, whichis therefore a line of curvature. In 
this case, therefore, the real: projections of the lines of curvature are 
the radii of this circle and all circles concentrical with it and in- 
cluded within it. 
If the equation of the surfaces which have a centre be ex- 
pressed in terms of their semiaxes it will be 
a°b’s* + a®c?’y? + b’c*a® = a®b’c® 
If the surface be an ellipsoid a2, b%, c*, will be all positive. 
If it be a double-surfaced hyperboloid a? and db’ are negative, and 
c’ positive. 
If it be a single-surfaced hyperboloid a? and Bb” are positive, and c* 
negative. 
The equation of the surfaces which have no centre may be ex- 
pressed 
y? + ma? = ps 
where m expresses the ratio of the squares of the axes of all elliptic 
or hyperbolic. sections perpendicular to the axis, and p the para- 
meter of the parabolic section of the surface by the plane yz. 
Since therefore for the surfaces having a centre 
‘ 
axah? a’ =a’? a” =b%, r= —ab’c” 
it follows that 
cits iy Rely (ne) 
& Tq aay oil @ (b? — c) 
| atta) gen 
a RU ik) —¢ 
