A RIGID BODY IY ELLIPTIC SPACE. 
297 
Again multiplying the expressions for x Q , y 0 , z 0 , u 0 by l v l 2 , 1 3 , 1 4 , respectively, and 
adding, we get, by virtue of these relations, 
x=l l x (} -\- Uy (j + J 3 z 0 + / A u (j , 
and there are similar expressions for y, z, u. 
If we square and add the new set of equations, and make 1 for all 
values of x 0 , y Q , z 0 , u 0 , we find another set of relations among the coefficients, of the 
types 
I* +Wi 3 +% 3 +Pi 3 =1 
/14 +m +n pin -\- r p — 0. 
We may include all these equations in a scheme similar to that used for orthogonal 
transformation in ordinary geometry 
X 
y 
z 
u 
a?o 
h 
n i 
Pi 
Z/o 
k 
m 2 
n 2 
Pz 
z o 
k 
m 3 
n 3 
Ps 
w 0 
h 
ni 4 
n i 
2h 
In this scheme, the sum of the squares of any row or column of the determinant is 
unity ; and the sums of the products of the corresponding terms in any two rows, or 
in any two columns, is zero. 
The square of the determinant by means of these relations reduces to unity, so that 
A a =l. 
If the positive directions of the edges of the tetrahedron retain the same relative 
positions towards each other, so that the tetrahedron could be moved into its new 
position, we must take A = + l. This may be easily verified for simple cases. This 
is the only case that concerns us in the motion of a rigid body. 
Comparing the equations 
Ax 0 =XqL a +y.jLo+" qL. 3 +p (J L J( 
X=1 A X o +hjy 0 +ky Q +hp 0 J 
where L 1? L 3 , L 3 , L^, are the minors of l v l 2 , l 3 , we see that each constituent in the 
determinant A is equal to its minor. 
25. Let the coordinates of the edges 23, 31, 12, 14, 24, 34 be (a, b, c,f g, h) 1 , 2 , 3 , 4 , 5 , 6 ' 
From the transformation of x 0 , y Q , z 0 , u 0 it is easily seen that 
Wo-!/o\) = (!/ z '-^)K )l 3 -¥ 2 )+ ... to six terms, 
MDCCCLXXX1V. 2 Q 
