236] FREEDOM OF THE FIFTH ORDER. 255
Then if 0 be a screw of the five-system with direction cosines cos X,
cos fi, cos v, and if x, y, z be a point on the screw 6 and p0 its pitch we must
have (§ 216)
(pv 4- a) cos X + z cos p — y cos v = 0.
Fig. 41.
Everything that we wish to specify about the five-system may be
conveniently inferred from this equation.
For example, let it be desired to find the locus of the screws of a five-
system which can be drawn through a given point x, y', z' and have the
given pitch .
We have (^fl+a) cosX+ /cos/4 — 3/cosi> = 0.
If x, y, z be a point on 0 we may substitute x' — x, y' — y, z' — z for cos X,
cos fi, cos v, and we obtain
(Pe + a) O' ~ (/ - y) - y' - 0) = 0,
whence we see that the locus is a plane, as has been already proved other-
wise. (When the pitch is zero, this is Möbius’ theorem, § 110.)
If we change the origin to some other point P which may with complete
generality be that point whose co-ordinates are o, h, o and call X, Y, Z the
co-ordinates with these new axes, the equation becomes
+ a)cos X + Zcos fi — (F + A) cos v — 0.