290 
MR. R. S. HEATH OH THE DYNAMICS OF 
and then (/, m, n, p ) will be regarded indifferently as the coordinates of the pole, or 
the coordinates of the plane. The distance of a point from any point of its polar plane 
is a right angle; and from what was proved for two dimensions it follows that any line 
passing through the pole of a plane is perpendicular to the plane. The lengths of the 
six edges and the angles of all the faces of the fundamental tetrahedron are all right 
angles. Such a tetrahedron is called a quadrantal tetrahedron. The coordinates 
( x , y, z, u) are the sines of the perpendicular distances of a point from the four planes 
of reference. If we put u — 0 in any formula the system reduces to the same coordi¬ 
nates as were used in two dimensions. Let m denote the sine of the perpendicular 
from any point (x, y , z, u) to the plane 
lx -f my -\-nz -\-p u — 0, 
then T3- is the cosine of the distance between ( x , y, z, u) and the pole of the plane, 
and therefore 
zj=Ix-\- my -b nz -f pu. 
The angle between two planes is equal to the distance between their poles, so that 
if 6 be the angle between the two planes (/, m, n, p), ( l ', m, n, p), 
cos 6=11' -\-mm -{-nn -{-pp. 
15. A straight line may be conveniently specified by six coordinates, as shown 
by Professor Cayley. Let (x, y, z, u), (pc', y , z, u) be two points on any line, and 
(/, m, ii, p), (V , in, n , p') two planes through it, so that 
lx -\-my -\-pii — 0^ 
Vx -\-m'y -{-nz -{-pu — 0 I 
lx -{-my -{-nz -{-pu =0 
I'x -{-my -{-nz -{-pu = 0 
Eliminating l between the first and third equations we get 
0 -j- m(xy — x'y ) — n(zx — z'x) -{-p(xu' — x'u) = 0 
Similarly, 
0+ m'(xy'—x'y ) — n'(zx' — z'x ) + p'(xu '— x'u) — 0, 
and we can obtain other equations of similar forms in the same way. Let a, b, c, J, g, h 
denote respectively the quantities 
