A Treatise on the Theory of Screws

298 THE THEORY OF SCREWS. [280- Unless in this exceptional case where is infinite it is always true that when pa is zero, a and t) are at right angles. It is universally true that when the impulsive screw and the instan- taneous screw are at right angles (the body being quite free), the pitch of the instantaneous screw must be zero. For if were not zero when cos(a^) was zero then å must be zero. As some motion must result from the impulse (the mass of the body being finite) we must have pa infinite. The initial motion is thus a translation. Therefore the impulse must have been merely a force through the centre of gravity ; a and 77 must be parallel and cos (a>?) could not be zero. The expression for the kinetic energy in § 279, assumes an indeterminate form when the impulsive wrench reduces to a couple. For we then have pa = 0, but as cos (a?;) is not zero the expression for mari, i.e. 4 {(p* + Pv) C0S (ai?) - sin (a7?)]> becomes infinite. The expression for the kinetic energy arising from an impulsive wrench of unit intensity on a screw ■»; applied to a free body of unit mass which thereupon begins to twist with an instantaneous movement about a screw a has the concise form cos (arj) ^0.7]’ Pa 281. Conditions to be fulfilled by two pairs of Impulsive and Instantaneous Screws. Let a be a screw about which a free rigid body is made to twist in consequence of an impulsive wrench administered on some other screw ry. Let ß be another instantaneous screw corresponding in like manner to £ as an impulsive screw. Then we have to prove that the two following formulae are satisfied*: —cos (/fy) + cos (a£) = cos (oh?) cos(/3f) Pa _ pß cos (a?/) cos (ߣ) * Proceedings of the Camb. Phil. Soo., Vol. ix. Part iii. p. 193.