A RIGID BODY IN ELLIPTIC SPACE. 
293 
therefore 
TiJ ^ t n 1 . — hy-\-gz—cm j 
rzgri — tz rvm= /;»: . —fz — bu i 
r ' y 
zsgi —=— gx-\-fz . —cu 
zspp — 777 zP = + . 
If we square and add these equations, the left side becomes TTf'-j-ard; hence 
cr 2 = a z (x z +« 2 ) -J- b 2 (y 2 -\- u 2 ) + c 2 (z 2 + u 2 ) -f/ 3 (y 2 + « 2 ) d-f7 2 (2 2 d“ * 2 ) + /t 2 (ad + ?/) 
-f- 2yz(bc — gh)-\-2zx (ca —hf)-\-2xy(ab—fg) 
-f- 2xu(cg — bh) -}- 2yu [ah — cf) -j- 2zu (bf — ag). 
18. Any line which meets a given line and its polar, cuts both perpendicularly, and 
the length of the part intercepted between them is a right angle. If’ we have two 
given lines in space, which do not meet, we can, in general, draw two lines cutting 
both perpendicularly; these are the two lines which can be drawn meeting the two 
given lines and their polars. These two common perpendiculars are conjugate to eacli 
other with reference to the absolute. 
Let (a, b, c,f g, h), (cV, b', g', hf) be two given non-intersecting lines, and let S 
be the length of one of their common perpendiculars. Draw a plane (l, to, n,p) through 
8 and the first line, and another plane (V, to', n', p) through § and the second line, and 
let 6 be the angle between these planes. Draw also a plane (X, /x, v, zs) through the 
first line and perpendicular to the plane (l, to, n, p), and (X', //, v, ts') through the 
second line and perpendicular to the plane (V, m , n',p'). Then we have the following 
relations:— 
/X +TO/x -\-nv -\-pzx = 0~j 
1\' -\-mp -\-nv -ppm' =0 I 
I'X -\-m p -\-n'v -pp'rz =0 I 
V\' -(- in'g - 1- n'v -\-‘p'tz' = 0 j 
IV +TOTO' '+ nn' -\-pp' — cos 6 
XX-f ~ gg + vv + COS S 
Now 
a =mv —/in, &c. 
toV— gri, &c. 
Hence it follows that 
act'd- W+cc +ff+gg'+ hh' 
= («'+ toto'4- nn +pp')(\\'+fifi'-\- vv trrsr ) 
— (IX-\-nv p-\-n'v-\-p'zs), 
