296 
MR. R. S. HEATH OH THE DYNAMICS OF 
follows, by Spherical Trigonometry, that the point describes a circle, whose centre is 
the foot of the perpendicular let fall from the point to the axis of rotation. In the 
case in which the point lies on the polar line of the axis of rotation, the radius of the 
circle is a right angle, or in other words the circle becomes a straight line, viz., the 
polar line itself; so that any point on the polar line of the axis of rotation remains 
on that polar line. 
A displacement which leaves all the planes through a given line unchanged in 
position is called a translation along that line. In such a displacement the poles of 
these fixed planes will be fixed points; in other words, all the points of the polar line 
are fixed. Hence a translation along any line is also a rotation about the polar line. 
If we measure a translation by the distance through which any point of the line of 
translation is moved, and a rotation by the angle through which any plane through 
the axis of rotation is turned, we see that a translation along any line is exactly the 
same thing as an equal rotation about the polar line. 
In elliptic space a translation has a definite line associated with it, just in the same 
way that a rotation has a definite axis. A translation through four right angles 
brings a body back to its original position. 
24. In working out the kinematics of a rigid body, we shall suppose a quadrantal 
tetrahedron fixed in the body, so that the whole theory will depend on orthogonal 
transformation. Let (l, m, n, y>) L % 3i 4 be the coordinates of the four angular points of 
the quadrantal tetrahedron moving with the body, referred to a fixed quadrantal tetra¬ 
hedron in space. Let ( x , y, z, u) be the coordinates of a point referred to the fixed 
tetrahedron, (x Q , y Q , z 0 , u 0 ) the coordinates of the same point referred to the other 
tetrahedron. 
Then 
x 0 =I ] x-\-m 1 y-\-n 1 z-\-p 1 u 1 
y 0 = l 2 x -f- m. 2 y + n 2 z -\-p 2 u I 
A= h x + m z>y + n z z +lh u j 
u 0 = lpc+myy + np +pyi j 
If we square these equations and add them and make 
O | O | 9 i 9 i 
X 0~ + Vq' + V + U 0~ — 1 
for all values of (x, y, z, u), we find relations among the coefficients, of the types 
1 1 3 +/ 3 2 + 4 2 + L 3 =1 I 
1 pw 1 +l. 2 m 2 +l d m. 6 -f 0 j 
