286] DEVELOPMENTS OF THE DYNAMICAL THEORY. 305
Multiplying similarly by/jj/Si, ...pnßn, and adding,
H u> qß ~~ •
Eliminating H, we find
WaaS’øi) — Ua1 +pßj-, ... 9n = Xan +pßn.
Substituting these in the two previous equations, we get
4- p^^ß = 0 ;
Xuaa + puaß = 0 ;
whence, as before,
Uaat^ßv ~ Uapttfay = 0.
286. Twist Velocity acquired by an Impulse.
From the fact that the twist velocity ä acquired by a free body in
consequence of an impulsive wrench of unit intensity on a screw y is
expressed (§ 280) by the equation
cos (at))
a —--------
Pa
we see that the second of the two formulae of § 281 may be expressed thus:—
ß^ßn =
The proof thus given of this expression has assumed that the body is quite
free.
It is however a remarkable fact that this formula holds good whatever
be the constraints to which the body is submitted. If the body receive the
unit impulsive wrench on a screw t), the body will commence to twist about
b. 20