A Treatise on the Theory of Screws

40] SCREW CO-ORDINATES. 37 and four similar equations; hence pnpn is proportional to the determinant obtained by omitting the reth column from the matrix or: a,, a3, a3, a4, a6, a8, A, A, ft, Ä, ä, T1> ?2> ?3> 74> 7s> 7«> • Si, 8.,, S4, 86, 56) €2> ^3) €4) e5) C6> and affixing a proper sign. The ratios of plt ... pe, being thus found, the actual values are given by § 35. If there were a sixth screw £ the evanescence of the determinant which written in the usual notation is (alt ß.2, ys, 84) e6, Q would express that the six screws had a common reciprocal. This is an important case in view of future developments. 40. Co-ordinates of a Screw on a Cylindroid. We may define the screw 0 on the cylindroid by the angle I, which it makes with a, one of the two principal screws a and ß. Since a wrench of unit intensity on 0 has components of intensities cos I and sin I on a and ß (§ 14), and since each of these components may be resolved into six wrenches on any six co-reciprocal screws, we must have (§ 34) 0n = a(l cos I + ßn sin I. From this expression we can find the pitch of 0: for we have pg = («1 cos 1 + ßi sin I)2, whence expanding and observing that as a and ß are reciprocal 'Slp1a1ß1 = 0, and also that ^,p1a1!1 = pa and 2p1/313=j0ß, we have the expression already given (§ 18), viz. Pe = Pa cos21 +pß sin21. If two screws, 0 and </>, upon the cylindroid, are reciprocal, then (m being the defining angle of </>), Sp, («i cos I + ßi sin l) (äj cos m + ß, sin m) = 0, or pa cos I cos m+ pß sin I sin m = 0. Comparing this with § 20, we have the following useful theorem :— Any two reciprocal screws on a cylindroid are parallel to conjugate diameters of the pitch conic. Since the sum of the squares of two conjugate diameters in an ellipse is constant, we obtain the important result that the sum of the reciprocals of the pitches of two reciprocal screws on a cylindroid is constant *. * Compare Octonions, p. 190, by Alex. M“Aulay, 1898.