for the equation to the trace on xy, which is therefore an 
ellipse AB; also, from the mode of generation, all sections 
parallel to coy are ellipses, and since their axes are in the 
ratio of a to h, they are all similar to AB. 
If we make x — h, we have 
y 2 z 2 h 2 
f- + - = 1 
Zr <r a 
for the equation to any section parallel to yz, which is an 
ellipse similar to the trace BC (since its axes are in the 
ratio of b to c whatever be the value of h) and which becomes 
imaginary when h>a. In the same manner it may be shewn, 
that all sections parallel to zx are ellipses, similar to the 
trace AC. The ellipses, of which AB, BC, CA are quadrants, 
in which the surface is intersected by the co-ordinate planes, 
which are its principal planes, are called the principal sections 
of the surface; and the parts of the co-ordinate axes inter 
cepted within the surface, are called its axes. Hence a, h, c 
represent the 1 axes of the ellipsoid, and also the 1 axes of 
its principal sections. The extremities of the axes such as 
A, B, C are called the vertices of the ellipsoid, of these it has 
six, one at the extremities of each axis. 
The whole surface consists of eight portions precisely 
similar and equal to that represented in the figure. 
Cor. If a — b, the equation becomes that to a spheroid 
generated by revolution about Oz ; similarly, if any other two 
of the semiaxes become equal, the ellipsoid becomes a spheroid 
generated by revolution about the remaining axis. 
59. To find the equation to the surface of a hyper 
boloid of one sheet. 
This surface is generated by a variable ellipse which 
moves parallel to itself, with its axes in two fixed planes, 
and vertices on two hyperbolas in those planes having a 
common conjugate axis coincident with the intersection of 
the planes.