420] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 465
In general let JTj = 0,
then cos 8 = ?3 ;
2 Vp3p4
whence we find that all objects in the extent X1 are displaced through equal
intervenes. This intervene can be readily determined for
e* + e-*= /Pl+ M
v Pi V p3
whence
or
8 = ^(log p3-log Pi)-
The intervene through which every object on X2 is conveyed has the
same value.
We could have also proved otherwise that objects on Xj and X2 are all
displaced through equal intervenes, for the locus of objects so displaced is
a quadric of the form
X\ X2 + X-X3^4 = 0,
and, of course, for a special value of the distance this quadric becomes
simply
W2 = 0.
If X1 = 0, and X3 = 0, then cos8 becomes indeterminate; but this is as
it should be, because all objects on X4 and X> are at infinity.
420. The Orthogonal Transformation*,
The formulae
yi = (11) «1 + (12) + (13) «s 4-(14) «4,
y2 = (21) »J + (22) x2 + (23) x3 + (24) xit
y3 = (31) «j + (32) x2 + (33) x3 + (34) xit
y4 = (41) + (42) x2 + (43) x3 + (44) xit
denote the general type of transformation. The transformation is said to be
orthogonal if when x4, &c., are solved in terms of y4, &c. we obtain as follows:—
x, = (11) y, + (21) y2 + (31)3/3 + (41)
«2 = (12) 2/x + (22) y2 + (32) y3 + (42) y4)
x3 = (13) y4 + (23) y2 + (33) y3 + (43) yit
«4 = (14) + (24) y2 + (34) y3 + (44) y4.
This is employed in Professor Heath’s memoir, cited on p. 452,
B.
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