426 THE THEORY OF SCREWS.
We notice here the somewhat remarkable circumstance, that if
[388-
(2 - «0P1 = 0, and Z22 + y0p2 = 0,
then all the screws on the cylindroid are permanent screws.
It hence appears that if two screws on a cylindroid are permanent, then
every screw on the cylindroid is permanent.
389. Three Degrees of Freedom.
Let us now specially consider the case of a rigid body which has freedom
of the third order. On account of the evanescence of the emanant we have
a dT a dT_ . dT
01 de;+ 0id.e; + 03 de;
o.
It is well known that if U, V, W be three conics whose equations submit to
the condition
xU + yV+ zW =0,
those conics must have three common intersections.
It therefore follows that the three equations
dT-0 -o ^-0
de; ' de; ’ de; ’
must have three common screws. These are, of course, the permanent
screws, and, accordingly, we have the theorem:—
A rigid system which has freedom of the third order has, in general, three
permanent screws.
There will be a special convenience in taking these three screws as the
screws of reference. We shall use the plane representation of the three-
system, and the equations of the conics will be
+ BfÅ + CfÅ = 0, or U=0,
A2é2é3 + = o, „ f=o,
j430203 + B303&i + CfhØz = 0, „ W = 0;
but, as éfJ + é2V4- 03W = 0,
identically, we must have
B, = 0; A2 = 0; A3 = 0 ;
C, = 0; C3 = 0; B3 = 0;
and also A± + B2 + C3 = 0.
For symmetry we may write
A1 = p — v, B2—v — X\ C3 = X —g.