30 DR. E. S. BALL’S RESEARCHES IN THE DYNAMICS OE A RIGID 
The condition of reciprocity is (art. 1) 
cos Q — d sin 3=0. 
Substitute 
a=p-\-m cos 23„ 
h=p-\-m cos 23 2 , 
d—m sin 20 1 —m sin 23 2 , 
3=t-(3,-3 2 ), 
we readily deduce 
tan 0, tan 
which proves the theorem (Salmon, 4 Conics,’ p. 152, 3rd edition) : — 
The asymptotes of the pitch hyperbola are parallel to the lines of zero pitch on the 
cylindroid. If p > m the hyperbola changes into an ellipse. In this case there are no 
screws of zero pitch on the cylindroid. 
30. On the ellipse of equal Jcinetic energy for a body possessing two degrees of freedom. 
— The pitch hyperbola is of merely kinematical significance ; the ellipse now to be 
described involves the conceptions of kinetics. 
Let 0 be a screw upon the cylindroid and co be the twist velocity about that screw ; 
u can be resolved into twist velocities a l and ao 2 about any selected pair of screws 3 15 3 2 
upon the cylindroid. The velocity of any element of the rigid body can therefore be 
expressed as a linear function of w l and &> 2 . The kinetic energy of the rigid body must 
therefore have the form 
-j- 2J5cu l ct) 2 T" Ca> 2 . 
In the principal plane of the cylindroid draw through the centre two lines parallel to 
the generators 3, and 3 2 , and with these two lines as axes construct the ellipse 
Ax 2 -j- 2 ~Qxy + Cy 2 = MA 4 . 
If § be the radius vector of the ellipse parallel to 3, 
CO CO | co 0 
q~x — y * 
whence the kinetic energy is 
MW 
We have thus proved that the kinetic energy due to a given twist velocity about any 
screw is proportional to the inverse square of the parallel diameter in the ellipse, which 
we may call the ellipse of equal kinetic energy. Or suppose a given quantity of energy 
is to be imparted by an impulsive wrench, the twist velocity that can be produced about 
any screw is proportional to the parallel radius vector. 
If a pair of conjugate diameters of the ellipse of equal kinetic energy had been taken 
as axes, the expression for the kinetic energy becomes 
Aa i -{- C u\. 
