A RIGID BODY IN ELLIPTIC SPACE. 
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and a simultaneous translation along it. Instead of the line we might equally well 
have used its polar. 
28. Suppose the position of any rigid body determined by the coordinates of the 
angular points of a quadrantal tetrahedron fixed in the body. Then, for a displacement 
of the body, we have the equations 
x — l x x o +l 2 y Q +4 2 o +4 w o ~j 
y=m 1 x Q +m. 2 y 0 +m 3 z 0 +m 4 w 0 i 
2= n i%o + n zVo + + z o + n * u o ' 
u=p l x Q +p 2 y 0 +ih z o +k u 0 J 
The sixteen quantities 4> l 2 , &c., have relations among them, and it is possible to 
express them all in terms of six properly chosen variables. These new variables 
aq, co 2 , <a 3 , oq, co 5 , co 6 are thus defined :— 
0J i = liPi + 4^2 + 4^3 + 4Pr 
^ 3 = n iPi + >hP 2 + n s2h 
oKy—mpi , + m. 2 n 2 + m s n s + m 4 w 4 
« 6 =Vi +%4 +%4 
o) 6 =l 1 m 1 2 + 4 m 3 + 4 7n 4 
Since 
4Pi + hP% +4Ps+4+4= 0 &c - 
we immediately deduce six other equations by differentiation. These are 
~ M \ — l\V\ + 4+2 + 42b + 4+4 
— g> 2 =m 2 y> 2 +m 3 _p 3 -f mpp 
— ° J 3 =n iPi + n yP? J A^hPs + n 4+4 
— aq= m l n l + m 2 w 2 -j- m 3 m 3 -fm 4 m 4 
~ 0) 5 =n 1 l 1 +Va + w s4 + n 44 
— a)Q=i 1 m 1 + 4 W7 2 + 4 to 3 + 4 m 4 J 
And besides these, we have the four equations 
o=44 +44 +44 +44 
0=w 1 w 1 *T m 2 m 2 +m 3 m 3 -f-mpr< 4 
0 = rqnj + w 8 n 3 +n 3 n 3 +%w 4 
0=2 7 i+i ++ 2 P 2 ++ 3+3 ++i+r 
