A Treatise on the Theory of Screws

282] DEVELOPMENTS OF THE DYNAMICAL THEORY. 299 To demonstrate the first of these formulae. Expand the left-hand side and it becomes + —t {(Ä + &) (t/j + %) + (ft + Ä) (% + 7?,) + (& + ße) (v, +%)} COS ((XT)) + —7^ {(«! + a2) (ft + + («3 + a<) (ft + + («5 + a<>) (& + ft)}. COS ( p ) But, as already shown, 'Pa 'Pa ---5----r = + CWtj, ? V2 = - aa2, ■■■, cos (ay)--------------------------cos (ay) - ~77m & = - a&> ■ • • ’ cos(/3f) cos(/3£) whence, by substitution, the expression reduces to + a (ß, + ß,2) (ax - a2) + a (ax + a2) (& - ß2) + b(ß3 + ßi) (a3 - a4) + 6 (a3 + a4) (ß3 - ßt) + c(ß6 + ße) (as - a6) + c (a6 + a6) (ßs - ße) — 2aa4ß4 — 2aa,2ß.2 + 2ba3ß3 — 2ba4ß4 + 2ca6/3B - 2ca3ß6 = 2-ara(3. To prove the second formula it is only necessary to note that each side reduces to + a?a$4 + (Pa..ß., + b"a3ß3 + bsa4ß4 + c'XA + c2a6ße. It will be observed that these two theorems are quite independent of the particular screws of reference which have been chosen. 282. Conjugate Screws of Inertia. We have already made much use of the important principle that is implied in the existence of conjugate screws of inertia. If a be reciprocal to £ then must y be reciprocal to ß. This theorem implied the existence of some formula connecting and •srj3l). We see this formula to be _ Pa _ Pß cos (ay) cos (ߣ) We have now to show that if = 0, then must = 0. Let us endeavour to satisfy this equation when ^ßri is zero otherwise than by making zero. Let us make pa infinite, then will reduce to cos (a£) (for we may exclude the case in which p% is also infinite because in that case = 0, inasmuch as any two screws of infinite pitch are necessarily reciprocal).