A RIGID BODY IN ELLIPTIC SPACE. 
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21. The general equation of a quadric is a homogeneous equation of the second 
degree in x, y, z, u. By an orthogonal transformation we know that we can rid the 
equation of the products of (x, y, z, u ), and at the same time keep x 2 -\-y 2 -\-z 2 -\-u 2 
unchanged. The equation will then reduce to the form 
Ax 2 -J-By 3 + Cz 2 + Du 2 = 0. 
The tetrahedron of reference is the common self-conjugate tetrahedron to the given 
quadric and the absolute. The six edges may be called the principal axes of the 
quadric. 
The equation to a system of confocal quadrics may, as before, be shown to be of the 
form 
x 2 if s 2 % 2 
a~ + A + 6 2 + X + c 2 + X^"^ 3 + X =0 ' 
Hence confocal quadrics cut at right angles. 
The kinematics of a rigid body. 
22. By a rigid body we mean a collection of particles so bound together that the 
distance between any pair of them remains the same, however the system be moved 
in space. In general, if we assume an arbitrary system of measure-relations as the 
basis of our definition of distance, a rigid body could not exist. But it is pointed out 
by Riemann in his paper “ On the Hypotheses which lie at the Bases of Geometry,” 
that the special character of those centinua, whose curvature is constant, is that 
figures may be moved in them from one position to another without stretching. This 
may be illustrated for two dimensions, by saying that any figure traced on a spherical 
surface may be moved from one position to another on the surface without deforma¬ 
tion. But on the other hand, a figure traced on the surface of an ellipsoid, or other 
surface for which the curvature is not uniform, can exist in one position only. Now 
in elliptic space there is a uniform positive curvature; hence we assume that a figure 
which exists in one position can exist in any other position of space without changing 
the distance between any two of its points. The same result is arrived at by Klein, 
by showing the possibility of finding a linear transformation, which transforms the 
absolute quadric into itself. 
23. A point is always distant a right angle from any point of its polar plane. 
Hence a point and its polar plane always move together like a rigid body. 
A displacement which leaves all the points of a given line unchanged in position, is 
called a rotation about that line. If we take any two fixed points on the line, the 
distance ol any point of the body from each of these remains unchanged. Hence it 
