301] DEVELOPMENTS OF THE DYNAMICAL THEORY. 321
Let a be the instantaneous screw and da the length of the perpendicular
thereon from the centre of gravity. If cos X, cos p, cos v be the direction
cosines of cZa then
cZtt cos X = (a5 — a6) (as + a4) c — (a5 + a6) (a3 — a4) b,
da cos p = (ax — a2) (a6 + a6) a — (a> + a2) (a6 — a6) c,
da cos v = (a3 - a4) (eq + a2) b - (a3 + a4) (a, - a2) a.
But if y is the impulsive screw corresponding to a as the instantaneous
screw we have
a«! = ——«ifc; — aa.2 = < %, &c-> &c.,
cos (ay) cos(cw7)
whence
da cos X = —~Z\ ((y,s + ye) (ys + ai) — («5 + a6) (ys + ^4))>
cos (ay)
da cos p = P*—r ((ifr + %) («5 + «») - (“i + «2) (% + %))>
cos (ay)
da cos v = —((’?s + Vt) («1 + «2) ~ (as + Mi) <d)i + Vid-
eos (ay)
But
(% + Vs) (<*s + «4) ~ («5 + ««) (Vs + O = sin (ay) cos V,
with similar expressions for sin (a?;) cos and sin (a??) cos / where cos X',
cos p, and cos v' are the direction cosines of the common perpendicular to a
and y. We have therefore
da cos X = —P, ~ \ sin (ay) cos X',
cos (ay)
^co^ = c^)sin(a’?)COS/A/)
da cos v = —P* , sin (ay) cos /,
cos (ay)
whence
cos X = cos X'; cos p = cos p ; cos v = cos v ;
and da = pa tan (ay),
which proves the theorems.
B.
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