A Treatise on the Theory of Screws

■MMMI 70 ______ THE THEORY OF SCREWS. [79 79. Screws of Reference. We have now to define the group of six co-reciprocal screws (§ 31) which are peculiarly adapted to serve as the screws of reference in Kinetic investi- gations. Let 0 be the centre of inertia of the rigid body, and let OA, OB, 00 be the three principal axes through 0, while a, b, c are the corresponding radii of gyration. Then two screws along OA, viz. — a; two screws OC, viz. ft>6, ®6) we shall employ. ft>i, w2, with pitches 4- a, along OB, viz. tö3, co,, with pitches + b, -b, and two along with pitches + c, — c, arc the co-reciprocal group which 1 he group thus indicated form of course a set of canonical co-reciprocals (§ 41). For convenience in writing the formulae, we shall often use p,, ... ps, to denote the pitches as before. We shall first prove that the six screws thus defined are the principal screws of inertia of the rigid body when perfectly free. Let the mass of the body be 21/, and let a great constant wrench on cu, act for a short time e. The intensity of this wionch is co, , and the moment of the couple is a®/'. We now consider the effect of the two portions of the wrench separately. The effect of the force is to give the body a velocity of translation parallel to OA and equal to ß Jl The effect of the couple is to impart an angular velocity about the axis OA. This angular velocity is easily determined. The effective force which must have acted upon a particle dm at a perpendicular distance r from OA is —dm. The sum of the moments of all these forces is J/re -- • This quantity is equal to the moment of the given couple so that Jlla2 = am,", e whence e „ = • 1 he total effect of the wrench on Wj is, therefore, to give the body a velocity of translation parallel to OA, and equal to J®", and also a velocity of rotation about OA equal to —. «/'. These movements unite to form a twisting motion about a screw on OA, of which the pitch, found by dividing the velocity of translation by the velocity of rotation, is equal to a. This same quantity is however the pitch of co„ and thus it is proved that an impulsive wrench on co, will make the body commence to twist about co,. We shall in future represent eco," by the symbol co,'", which is accordingly to express the intensity <j the impulsive wrench.