130 THE THEORY OF SCREWS. [142-
Let X, Y (Fig. 23) be the two principal screws of inertia. Let A be
an impulsive screw, and A' the corresponding instantaneous screw. Draw
Fig. 23.
through A the line AH parallel to XY. Join HA', and produce it to
meet the homographic axis at fl. Let a be the twist velocity generated by
an impulsive wrench of unit intensity at X, and let ß be the corresponding
quantity for Y.
It may be easily shown that the triangle AA'X is similar to YA'n, and
that the triangle AA'Y is similar to XA'n; whence we obtain
A'X__nÆ A'Y nA'
AX “OF’ AY~nX •
The unit wrench on A can be decomposed into components on X and Y of
respective intensities
AY AX
XY’ XY'
These will generate twist velocities
4F „AX
“XY’ XY'
Let to be the resulting twist velocity on A', then the components on X and F
must be equal to the quantities just written ; whence
A'Y _ AY
“ XY~a XY’
A'X AX
“ XY~^XY’
and we obtain
nA' n nA'
a~a} nx; nY ’
or, a : ß :: (1Y : nX;
we thus see that 11 is a fixed point wherever A and A' may be.