20 
DR. R. S. BALL’S RESEARCHES IN THE DYNAMICS OF A RIGID 
Ag, the only operative wrenches are those about A„ A 2 , for all the others are reciprocal 
to P, and are destroyed by the reaction of the constraints.. Hence the wrench X x must 
he such as to neutralize the effect of the wrench X 2 in its efforts to disturb the equili- 
brium of a body only free to twist about P. Therefore the wrenches X 15 X 2 must be 
such as compound into a wrench about the screw on the cylindroid reciprocal to P. Thus 
the ratio of Xj to X 2 is completely determined. 
9. Expressions for the components about six coreciprocal screws into which any wrench 
may be decomposed. — The use of the virtual coefficient will afford concise values of the 
components. Let A 1 See., A 6 be the coreciprocal screws, and let X be the wrench about 
the screw S which is to be decomposed. Let^? m be the pitch of the screw A m . Let K OT 
be the virtual coefficient of S and A m . Let X m be the component wrench about A m . 
The energy expended in giving a body a small twist a around A, against the wrench X is 
aXRj ; 
this must equal the energy expended in giving the body the same twist in opposition to 
the component wrench X 1 ; for since A 2 &c., A 6 are reciprocal to A„ the wrenches 
X 2 &c., X 6 cannot affect the quantity of energy required. The virtual coefficient of 
two coincident screws reduces to the sum of the pitches, and therefore 
&XR.! = 2<yX]2?], 
whence 
x '= x |- 
The analogy of this process to the resolution of a force or velocity along three rectangular 
axes may be noticed. The velocity of a point is resolved into three components parallel 
to the axes by multiplying the velocity into the direction cosines ; so the twist velo- 
city of the rigid body is resolved into six twist velocities about six coreciprocal screws 
by multiplying the original twist velocity into six functions analogous to the cosines. 
10. Relation between the square of a wrench and the squares of its components along 
a coreciprocal system. — We can also detect a theorem analogous to the familiar truth 
that the sum of the squares of the three component velocities is equal to the square of 
the original velocity. 
To twist a body through a small angle a about the screw S, in opposition to the 
wrench X about the same screw, requires a quantity of energy equal to 
2<w .p .X; 
but this must equal the quantity of energy necessary to overcome the components, 
namely, 
^(XjRj + & c. + XgRg), 
but (art. 9) 
substituting, we find 
pA =i?iX* -f- &c. +p 6 xi 
