12 
DE. E. S. BALL’S EESEAECHES IN THE DYNAMICS ON A EIGID 
Let S be any other screw of pitchy coordinate with the system just written. 
Let X be the magnitude of a wrench about S which is decomposed into wrenches 
Xj &c., Xj. about the Jc screws A x &c., A*. 
Giving the body a small twist a about X re , and denoting by R m> „ the virtual coefficient 
of A m , A„, we have the following equations, art. (10) : — 
XR 1>s =2p ] X 1 +R 1>2 X 2 + &c.+R 1 ; 7 c X„ 
XR 2(S — 2R 2jl X 1 +2p 2 X 2 + &c. R 2; iX ft , 
XR £jS =2R fcil X 1 +R, )2 X 3 +&c.+2pA. 
But giving the body a small twist a about S, we have 
2y>X=R Sj jXj -f- R s , 2 X 2 + & c - + 6^6. 
Eliminating R s> x &c., we have, finally, 
i? X 2 = i?1 X?+& C.+&KI+SB*, „X m X B . 
For six coreciprocal screws this result of course reduces to the relation of art. (10). 
33. Construction of the principal screws of inertia for a body with three degrees of 
freedom. — We have demonstrated [art. 88] that all the screws parallel to a plane 
selected from the general system of three degrees of freedom lie on a cylindroid. A 
section of the pitch hyperboloid drawn through the kinematic centre gives the pitch 
hyperbola appropriate to the cylindroid. We hence infer the following theorem : — 
, Any set of three coreciprocal screws selected from the general system of three degrees 
of freedom must be parallel to three conjugate diameters of the pitch hyperboloid. 
The principal screws of inertia must therefore be parallel to three conjugate diameters 
of the pitch hyperboloid; but they must also be parallel to three conjugate diameters 
of the ellipsoid of equal kinetic energy [art. 88], and hence the principal screws are 
completely determined. 
YI. MISCELLANEOUS PEOPOSITIONS. 
34. On the locus of the displacements of a point which can he produced hy twists 
about the screws on a cylindroid. — Let P be a point and A, B be any two screws on a 
cylindroid. If the body to which P is attached receive a small twist about A, the 
point P will be moved to P'. If the body received a small twist about B, P would be 
moved to P". Then, whatever be the screw C on the cylindroid about which the body 
be twisted through a small angle, the point P will still be displaced in the plane PP'P". 
For the twist about C can be resolved into two twists about A and B, and therefore 
every displacement of P must be capable of being resolved along PP' and PP". 
Thus, through every point in space a locus plane can be drawn to which the small 
movements of that point arising from twists about the screws on a cylindroid are confined. 
The simplest construction for the locus plane is as follows : — Draw through the point 
P two planes, each containing one of the screws of zero pitch : the intersection of these 
planes is normal to the locus plane through P. 
