A RIGID BODY IN ELLIPTIC SPACE. 
303 
These equations give us 
4= • + <44-^4+<44 ] 
4=-<Vi • + <44+^4 l, 
4— <44-^4 • +<44 j 
4 — 44 44 44 • J 
and similar equations in w?, n, and p. 
It will be seen that — 0 l3 — 0 3 , — 0 3 , — 4? — <4> — 4> are formed from the coordinates 
(4, 4, 4, 4)> & c -> i n exactly the same way as aq, aq, w 3 , aq, oi 5 , cu 6 , were formed from 
(l, m, n, p), &c. Hence, if we regard the variable tetrahedron as fixed, the other 
tetrahedron will have angular velocities — 0 l5 — 0 3 , — 0 3 , — 0 4 , —0- o , — 0 6 relative to it. 
Hence, reversing the motion, we see that 0 4 , 0 3 , 0 3 , 0 4 , 0 5 , 0 G are the angular velocities 
of the moving tetrahedron about fixed axes instantaneously coinciding with its own. 
31. By definition, 
0i = hh+ m i' >1 h+n l n i -\-2hP±- 
Substitute the values of l x , m v n v p x in terms of the ofs; then 
^.e., 
4 = "1 {hlh—kPi )+ "a( m iP4 ~ m 4Pi )+"3 {ni2h ~ Vi) 
+ta 4 (TO 1 n 4 -«i 4 w 1 ) + <u 6 (« 1 4-nJ 1 )H-fi > 
— "l/l + "20fi+ W 3 4 + W 4 a 4+ W 6 C 4 
^1 — "i a i+ "A + "3 c i + "r/i + "s0'i+" 0 + 
Similarly for the other 0’s. Hence the 0’s can be expressed in terms of the ofs by 
the same transformation scheme that was used for expressing a 0 , b 0 , c Q ,f 0 , g 0 , h 0 , in terms 
of a, b, c,f g, h. 
32. This result gives us the law by which angular velocities and translations are 
resolved. For suppose all the 0’s zero except 0 4 , then 
(o ^ — CL 4 0 4 
ca 2 =b 1 0 1 > 
W 3= C 1 @1 . 
"4=/l4“ 
" 6 —9\0\ > 
In other words, any angular velocity 0, about a fixed line (a, b, c,f, g , h) is equivalent 
to component angular velocities «0, b9, c9,f9, g9, h9 about the six edges of the funda¬ 
mental tetrahedron respectively. 
A translation along a line may be resolved into translations along the six edges 
according to the same law. 
