422 THE THEORY OF SCREWS. [386-
Introducing the value just obtained for T,
Mufø1 = + 02(A01 + B0i),
MufØ^-Ø, (AØ. + BØf
There must be one screw on the cylindroid, for which
4^ + M = o.
This screw will have the accelerations öj and 02, both zero, and thus we have
the following theorem :—
If a rigid body has two degrees of freedom, then, among the screws about
which it is at liberty to twist, there is one, and in general only one, which has
the property of a permanent screw.
The existence of a single permanent screw in the case of freedom of the
second order seems a noteworthy point. The analogy here ceases between
the permanent screws and the principal screws of inertia. Of the latter
there are two on the cylindroid (§ 84).
387. Geometrical Investigation.
Let N (fig. 43) be the critical point on the circle which corresponds to
the permanent screw (§ 50). Let Pbe a screw 0, the twist velocity about
which is 0. Let u6 be a linear parameter appropriate to the screw 0, such
that Muq202 is the kinetic energy.
Let 0, and O2 be the two screws of reference on the cylindroid and for
convenience let the chord Off be unity. Let the point Q correspond to
another screw </>, then from §57
01=0PO2, 02=0PO1; <j>i == <j>Q02, <£2 = rfiQCp
Ptolemy’s theorem gives
PQ0^ = 0^ - 0^.
Now let be the adjacent screw about which the body is twisting in a time
8t after it was twisting about 0.