472
THE THEORY OF SCREWS.
[425-
with, as before, the condition,
a2 + ß2 + + 82 = 1.
If 0 be the intervene through which the vector displaces an object then
it is easily shown that cos 6 = a.
426. Parallel Vectors.
The several objects of a content are displaced by the same vector along
ranges which are said to be parallel.
Taking the space representation, § 413, Clifford showed that all right
vectors, which are parallel, intersect two generators of one system on the
infinite quadric, while all left vectors, which are parallel, intersect two
generators of the other system.
A generator intersected by two rays from a right vector maybe defined by
the points whose coordinates are
+ O-', — ß, ~ y, —5,
+ ß, + a', — 8, + y,
while a generator intersected by two rays from a left vector will be
defined by
+ “o', — ßo, — yo, — So,
+ ßo, + a0', + S», — 7o-
To prove the theorem, it is only necessary to show that these four points
are coplanar, for then the two generators intersect, i.e. are of opposite
systems. We have, then, only to show that the following determinant
vanishes:—
a — ß — y — 8
ß a — 8 y
«o' - ßo — 7o —
ßo a-o 8„ — yc,
This will be most readily shown by squaring, for with an obvious notation
it then reduces to the simple form
0 0 [4'] [a'A]
0 0 m]
[«'«o'] r/soo'j 0 0
[Ä“'] [M] 0 0
whence we see that the original determinant is simply
[«'«o'] [/3a0']
[&>«'] [M] ’