A RIGID BODY IN ELLIPTIC SPACE. 
291 
then 
yz'—y'z, zoT—z'x, xy'—x'y, xv!—xfu, yu'—y'u, zu'—z'u, 
. —cm-f bn—fp =0 j 
cl . — an—gp= 0 
-bl-\-an . — lip — 0 
fl-\-gm-}-hn . =0 
There is a similar set of equations obtained by writing V , m, n', p' for l, m, n, p in 
these equations. 
Taking the equations 
—cm -\-bn —fp =0 
—cm'-\-bn'—fp — 0 
and eliminating f we have 
c(mp' — mp) — b(np r — n'p) 
Proceeding in this way, and eliminating the other letters in turn, it is easily seen 
that 
a : b : c : f : g : li 
— lp—Vp : mp'—mp : np' — n'p : mri — ru'n : nl' — n'l : I'm —I'm. 
Choose the planes (l, m, n, p), ( l', m, n, p') to be at right angles, and the points 
(x, y, z, u), ( x, y, z\ u) to be distant a right angle ; then 
(Ip — I'pYf- (mp — m'pY(np' — n'p) 2 + (mri — m'n) 2 -f- (nl'—(lm — I'nif 
(Z' 3 -f m 2 + p 2 )((' z -\~ m' 2 + n' 2 -\-p' 2 ) — (IT + mm' + nn'-\-p p') 3 
= 1 , 
and similarly it may be shown that 
(yz'—y'z) 2 -\- (zx — z'x) 3 + (xy' — x'y) 3 + (xp —afp ) z -f (yp — ypf-\- (zp — z'p) - = I. 
Hence 
a = yz'—y'z—lp' — Vp, 
and so on for all the letters, and 
cr-f lr-\-c 2 -\-f' 2 -Tg 2 -{-h 2 = l. 
From the forms of a, b, c,f g, h, it is easy to show that there is an identical 
relation between them 
af-{-bg-\-ck=0. 
2 p 2 
