466
[420-
THE THEORY OF SCREWS.
From the first formulæ the equation for p is, as before, § 417
(ll)-p (12) (13) (14)
(21) (22) -p (23) (24)
(31) (32) (33) -p (34)
(41) (42) (43) (44) — p
From the second, the equation for p must be
W-p (21) (31) (41)
(12) (32) (42)
(13) (23) (33) - * P (43)
(14) (24) (34) (44) - -
P
but we may interchange rows and columns in a determinant so that the last
may be written,
(1D-1 (12) (13) (14)
(21) (22)-1 (23) (24)
(31) X32) (33)-- P (34)
(41) (42) (43) (44) - -
whence we see that the equation for p must be unaltered, if for p we sub-
1
statute -. It must therefore be a reciprocal equation of the type
p4 + 4 yip3 + GBp2 + 4,Ap + 1 = 0,
and the roots are of the form
and as
1.
’ p"'
'4=o,
p
, i
, i
this transformation fulfils the fundamental condition (§ 417).
421. Quadrics unaltered by the Orthogonal Transformation.
The special facilities of the orthogonal transformation in the present
subject arise from the circumstance that it is the nature of this transforma-
tion to leave unaltered a certain family of quadrics. This is as we have