354 THE THEORY OF SCREWS. [323
radii of gyration of the two bodies, around any one of these three diameters,
will not be equal. If, however, it should happen that the radius of gyration
of one body be equal to that of the other body about one OX of these three
common conjugate diameters, it can be shown that any screw, parallel to OX,
whatever be its pitch regarded as an instantaneous screw, will have the same
impulsive screw for either of the two bodies.
If the radii of gyration about two of the common conjugate diameters
were equal for the two bodies, it will then appear that any instantaneous
screw which is parallel to the plane of the common conjugate diameters, will
have the same impulsive screw for each of the two bodies. The corresponding
impulsive screws belong to the system of the fifth order, which is defined by
being reciprocal to a screw of zero pitch on the third of the three common
conjugate diameters.
Of course, if the radii of gyration coincide on this third diameter, then
the two rigid bodies, regarded from our present point of view, would be
identical.