425] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 471
From (iv), ßy = -(24) (34).
From (v), by multiplication, ßü = +(23)(34); ß^ = - (23) (24) (34/;
but, from (vi), 78 = - (23) (24);
whence, we deduce, /32=(34)2.
The significance of the double sign in the value of ß will be afterwards
apparent; for the present we take
ß = + (34).
From (ii) 5 = + (23),
From (iii) 7 = “ (24),
while the group (vii) will be satisfied if
a2 + ß2 + 72 + & = 1.
The scheme of orthogonal transformation for the Right V ector (for so we
designate the case of ß = + (34),) is as follows:—
+ a + ß + V + 8
~ß + ot + 8 - 7
~ 7 — 8 + a + ß
- 3 + 7 ~ß + a
If we append the condition
a? + ß2 + V2 + S2 = 1,
then we have completely defined the Right Vector.
We now take the other alternative,
/3 = - (34);
then, from (ii), 3 = — (23),
then, from (iii), 7 = + (^4).
We thus have for the Left Vector, the form,
+ a. + ß +7 +S
- ß + a -8 +7
— fy +2 + H ß
— 8 —7 + ß + ®