A Treatise on the Theory of Screws

97] THE PRINCIPAL SCREWS OE INERTIA. 85 97. Co-ordinates of Impulsive and Instantaneous Screws. Taking as screws of reference the n principal screws of inertia (§ 84), we re- quire to ascertain the relation between the co-ordinates of a reduced impulsive wrench and the co-ordinates of the corresponding instantaneous screw. If the co-ordinates of the reduced impulsive wrench are y"', ...i?»"', and those of the twist velocity are d1; d2)... ån, then, remembering the property of a principal screw of inertia (§ 78), and denoting by ult ... un, the values of the magnitude u (§ 89) for the principal screws of inertia, we have, from § 90, • ^l2 1 H> aiP1=MV1 ’ whence observing that å1 = åa1; ... = da„, we deduce the following theorem, which is the generalization of § 80. If a quiescent rigid body, which has freedom of the nth order, commence to twist about a screw a, of which the co-ordinates, with respect to the principal screws of inertia, are an and if plt ... pn be the pitches, and wi, ... un the constants defined, in § 89, of the principal screws of inertia, then the co-ordinates of the reduced impulsive wrench are proportional to Let T denote the kinetic energy of the body of mass M when animated by a twisting motion about the screw a, with a twist velocity a. Let the twist velocities of the components on any n conjugate screws of inertia be denoted by d1( d2,... an. [These screws will not be co-reciprocal unless in the special case where they are the principal screws of inertia.] It follows (§ 88) that the kinetic energy will be the sum of the n several kinetic energies due to each component twisting motion. Hence we have (§ 89) T = Mu^äi + ... + Mu„2ån2, and also wa2 = u2a2 + ... + un2a»2- Let a1; ... an and ßlt ... ßn be the co-ordinates of any two screws belong- ing to a screw system of the nth order, referred to any n conjugate screws of inertia, whether co-reciprocal or not, belonging to the same screw system, then the condition that a and ß should be conjugate screws of inertia is Wi2«iA + • • • + ‘Un«nßn = 0. To prove this, take the case of n = 4, and let A, B, C, D be the four screws of reference, and let A,, ... Ae be the co-ordinates of A with respect to the six principal screws of inertia of the body when free (§ 79). The unit wrench on a is to be resolved into four wrenches of intensities a,, ... a, ou A, B, C, D: each of these components is again to be resolved into six wrenches on the