458
THE THEORY OF SCREWS.
[417
We thus get
Hp1p2p3p4X1 — X1 x2 «3 x4 11 12 13 14
Xj x2 •21 22 23 24
x2" Xi" 31 32 33 34
x"" x2" ®3"" 41 42 43 44
= P2X1
y-3, p2^2
y-3, p2 ^3
Vi, p2^4
or HP1X^ yi
x-['
x("
x('"
But these determinants
tetrahedron
are the co-ordinates
, and omitting needless factors
/W", piOO,"''
pax."', p4x2""
Ps^s ) p4^3
p3^"', P4<"
y* ys V4
X2 00$ x±
x$ x±
™ 'Ht 1111 >'ll
^2 00$ 00^
of y referred to the new
1^1 Pl^l> 1 2 — p2-^2> 1^3 — Ps-^31 1'1 — p4^~4’
We thus obtain the following theorem.
Let a\, x2, xs, x4 be the co-ordinates of a point with respect to any arbitrary
tetrahedron of reference.
Let 2/i> 2/a, y3, 2/4 be the co-ordinates of the corresponding point in a
homographic system defined by the equations
2/1 — (11) æi + (12) x2 + (13) x3 + (14) a?4,
y2 = (21) «J 4- (22) x2 + (23) x3 + (24) x4,
y3 = (31) + (32) x2 + (33) x3 + (34) x4,
2/4 = (41) x4 + (42) x2 + (43) x3 + (44)
If we transform the tetrahedron of reference to the four double points of
the homography, and if Xlt X2, X3, X4 be the co-ordinates of any point
with regard to this new tetrahedron then the co-ordinates of its homographic
correspondent are
p1^-2> p3-^-3> Pi^4>
where plt p2, p3, p4 are the four roots of the equation,
(11) -p (12) (13) (14) = 0.
(21) (22) -p (23) (24)
(31) (32) (33) -p (34)
(41) (42) (43) (44) — p