A Treatise on the Theory of Screws

284] DEVELOPMENTS OF THE DYNAMICAL THEORY. 303 We thus obtain and similarly I /// i" rcos^wxø. z ß 1 z// a ’y jjrcos (a£) _ ■n whence ßf" cos (<) or remembering that pa is infinite, • J-/// 7?_ ^=^2Lß ßg" But we had already from (iii), ßt"' ™yi ’ whence we deduce that in this case also the formula remains true. We thus obtain the following general theorem. 7/’ y, %, be three impulsive screws and a, ß, y the three corresponding instantaneous screws, then in all cases, no matter how the movements of the body may be limited by constraints, the following formula holds good*-. It is easily shown that this relation subsists when the correspondence between y and a is of the more general type implied by the equations _\dU_______________________\dU Vl ~ p, da2 ’ p2 da2 ’ Ve pa da„ ’ where U is any homogeneous function of the second degree in the co- ordinates. 284. Case of a Constrained Rigid Body. Let y and £ be, as before, a pair of impulsive screws, and let a and ß be the corresponding pair of instantaneous screws. Let p be the screw on which a reaction is contributed by the constraints at the moment when the impulsive wrench is applied on y. The movement of the body twisting about a is therefore the same as if it had been free, and one impulsive wrench had been imparted about y and another simultaneously about p, so that the following conditions are satisfied: api«i = y yi + p pi> åp6ae = y'''y<> + p'"ps- * Trans. Roy. Irish Acad., Vol. xxx. p. 575 (1894).