A Treatise on the Theory of Screws

298] DEVELOPMENTS OF THE DYNAMICAL THEORY. 317 Let X be the screw on A which is reciprocal to P, e .....................P ............... A. Then any screw p, on A and any screw </> on P fulfil the conditions = 0, = 0. Hence is the impulsive screw corresponding to y as the instantaneous screw. 298. Three Pairs of Correspondents. Let a, y; ß, %', y, £ be three pairs of impulsive and instantaneous screws ; let 0, cj> be another pair. Then, if we denote by Laß = 0, and Maß = 0, the two fundamental equations —cos (ßy) d----------------^7ät\ 008 (a£) = 2'53’«ß> cos(a7?) cos(/3£) cos (ay) we shall obtain six equations of the type Lea = 0, Lep — 0, Mea = 0, Meß = 0, cos (ß%) Ley = 0, Mey = 0. From these six it might be thought that </>j, ... </>6 could be eliminated, and thus it would, at first sight, seem that there must be an equation for 0 to satisfy. It is, however, obvious that there can be no such condition, for 0 can of course be chosen arbitrarily. The fact is, that these equations have a peculiar character which precludes the ordinary algebraical inference. Since a, y; ß, £; y, £; are three pairs of screws, fulfilling the necessary six conditions, a rigid body can be adjusted to them so that they are respectively impulsive and instantaneous. We take the six principal screws of inertia of this body as the screws of reference. We thus have, where Pa, pß, Py are certain factors, pay1 = æ«i> payi = —aa2, pay3 = ba,, payi = — bait Pß^ = aßlt ................., ........., ............, py^i = ayi> ................, ..........> ............. By putting the co-ordinates in this form, we imply that they satisfy the six equations of condition above written. Substituting the co-ordinates in Lea = 0, we get 0 = +(<%! + a2) (pe </>! + pe </>2) + (a, + aß (pe </>s + pe </><) + (a5 + a6) (pe </>6 + pe </>,) + (0i + 0ß Oi - aa2) + (03 + 0ß (ba, - baß + (0, + 0ß (cas - caß - 2 (aa101 - aa,0ß - 2 (ba,03 - ba40ß - 2 (ca,0s - ca906).