A RIGID BODY IN ELLIPTIC SPACE. 
299 
For let a, b, c,f g, h be the coordinates of a line which remains unchanged. If 
such a line exist, we must have 
0 = a(a 1 — 
1 ) + &&! 
d-ccj 
+dj4 
+^1 
-f/i/zj 
0=aa 2 
+ &(& 3 “ 
l)+cc 2 
+/A 
+ hh 2 
0 = act 3 
+ 663 
1 
CO 
+ 
1 ) ~h//.3 
+ 5^3 
fi-Mg 
0=aa 4 , 
+ % 
+ CC 4 
+/(/ 4 “ 
!)+m 
-\-bh^ 
>0 
II 
0 
+ &&. 
+ cc 5 
+// 5 
+^(^ 5 - 
1)+M 5 
0 =aa 6 
+ 66 e 
+cc 6 
These equations are not independent; they are equivalent to four independent 
equations only. 
We proceed to eliminate g and li between the equations 2, 3, and 4. The deter¬ 
minants thus introduced can be simplified by virtue of the relations between 
{l, m, 71, p) h% -sA- Thus 
foA) =( m ?Ah - m i lh)A p, - n 2 p x ) - (m lft - mppj (w 3 p 1 - n x p 8 ) 
=p 1 {mAiP2- n 2Pi) + m 2( n 3Pi- n iPz)+ m i(n. 2 Pa-'>hp l )} 
— PiC 
Similarly 
(sA)—ih4 {9J 1 2 )— Pih- 
Again 
M 9-A) + <h(ffA) + a 3 {gji a ) 
= —Pi { k( m Ph ~ n h n 1 ) + k( m i n s— m Ph) + k( m i n 2 ~ m -2 n i )} 
— Pi{ m vh n i — 
Similarly 
Wa) + )(pA) J rh(9Jh) — — ihO'h + 4) 
c*(sA) + c o(y 3 /q) +(c 3 -i )(sA)= Pi{k+™ i) 
(/i-iXS'AR/afsA) +/s(fl r A) = Ai (4 +Ai). 
Hence (finally) 
—pM^i + 4) +i?ic(4+mj +Pi/(4+Pi) = 0 . 
By a similar process we arrive at the following equations 
+&(% +4) — c (4 + m i)~/(4 +Pi) = °"] 
—«( m 3+%) • +c(4 d-Wi)— P'(m 4 +p a )=:0 \ 
a(m 3 +7i 2 ) — b(7i 1 +4) . —/t(%+i> 3 ) = 0 | 
/(4 +i>i)+fl f (w 4 +i>a)+^K+^8) • =° j 
2 Q 2 
