24 
DE. E. S. BALL’S EESEAECHES IN THE DYNAMICS OF A EIOID 
We deduce from this the equations 
P=0, Q=0, R=0, 
two of which, being independent, provide the two conditions required. 
16. On a relation between a sextant and a virtual coefficient. — We shall first state a 
simple vector problem. Given three vectors a, 0, y, determine the cosine of the angle 
between y and the common perpendicular to a and 0. 
The required cosine is the quotient obtained by dividing the determinant whose 
evanescence shows a, 0, y to be coplanar by the sine of the angle between a and 0. 
The corresponding question in the theory of screws is as follows. Given six screws 
Aj &c., A 6 of which S represents the sexiant. Let B m be the screw reciprocal to the 
five screws A x &c., A m _ 1? A m+1 &c,, A e , and let R m be the virtual coefficient of A m 
and B m . Let K m denote the function 
\/ P 2 -f- Q 2 + R 2 , 
computed for the five screws A x See., A m _ x , A m+1 &c., A 6 . Then R m can only differ by 
a numerical factor from 
S_ 
Rm 
For R m must vanish if S vanish, unless at the same time K m vanish. S is of the third 
dimensions of linear magnitude and K TO of the' second, so that the quotient is of the 
same dimensions as R TO . 
IY. ON IMPULSIVE SCEEWS AND INSTANTANEOUS SCEEWS. 
17. The impulsive cylindroid and the instantaneous cylindroid. — A rigid body M is at 
rest in a position P, from which it is either partially or entirely free to move. If M 
receive an impulsive wrench about a screw X 15 it will commence to twist about an 
instantaneous screw A x . If, however, the impulsive wrench had been about X 2 or X 3 
(M being in either case at rest in the position P), the instantaneous screw would have 
been A 2 or A 3 . Then we have the following theorem : — 
If X 15 X 2 , X 3 lie upon a cylindroid S (which we may call the impulsive cylindroid), then 
A 1? A 2 , A 3 lie on a cylindroid S' (which we may call the instantaneous cylindroid) *. 
For if the three wrenches are of suitable magnitude they may equilibrate, since they 
are cocylindroidal ; when this is the case the three instantaneous twist velocities must 
of course neutralize, but this is only possible if the instantaneous screws be cocylindroidal. 
18. On an anharmonic property of the impulsive and instantaneous cylindroids. — If we 
draw a pencil of four lines through a point parallel to four generators of a cylindroid, 
the lines forming the pencil will lie in a plane. We may define the anharmonic ratio 
of four generators on a cylindroid to be the anharmonic ratio of the parallel pencil. We 
shall now prove the following theorem : — 
The anharmonic ratio of four screws on the impulsive cylindroid is equal to the 
anharmonic ratio of the four corresponding screws on the instantaneous cylindroid. 
* When three screws are contained on a cylindroid, the screws may, for brevity, be said to he cocylindroidal. 
