CHAPTER XXI.
DEVELOPMENTS OF THE DYNAMICAL THEORY.
279. Expression for the Kinetic Energy.
Let us suppose that a body of mass M is twisting around a screw a with
the twist velocity å. It is obvious that the kinetic energy of the body must
be the product of Ma? and some expression which has the dimensions of the
square of a linear magnitude. This expression has a particular geometrical
significance in the Theory of Screws, and the symbols of the theory afford a
representation of the expression in an extremely concise manner.
Let -7 be the impulsive screw which corresponds to a as an instantaneous
screw, the body being supposed to be perfectly unconstrained.
As usual pa is the pitch of a and (at)') is the angle between a and y.
From the formulæ of § 80 we have, where H is a common factor,
whence
Hy^ — A-aap, Hy.,= — aa2;
Hy3 = + ba3; HVi = - bap,
Hvs = + c«5; Hye — — cap,
H K1?! + %) (ai + a2) + (y3 +y4) (a3 + a4) + (y5 +y3) (a5 + a6)J
= a (a,2 - a./) + b (as2 - a42) + c (a62 - a62) = pa
and we obtain
cos(ay)
The kinetic energy is
Md1 (a2aj2 + asa22 + b2a32 + 62a/ + c2a62 + c2a(i2)
= ^a2 cos (ay) ^aa,V1 ~ aa2V2 + ~ + ca^b> ~ ca6->/6]
= Må? —-i tn*,
cos (ay) v
* Trans. Roy. Irish Acad., Vol. xxxi. p. 99 (1896).