A Treatise on the Theory of Screws

APPENDIX I. 495 of reference. Let a4, ... as and ß4, ... ße be the co-ordinates of a and ß with respect to A, B, G, &c. Let Alt ... be the co-ordinates of A with respect to the six principal Screws of Inertia and similarly let Blt ... Be be the co-ordinates of B and in like manner for G, 1), E, &c. An impulsive wrench on a of intensity a" will have for components a"'a, on A ... and a"'a6 on F. These components on A, ... F may each be resolved into six component wrenches on the principal screws of inertia, viz. a"a1A, + a"a2B4 ... + a"a6F1, m A .>>> T) . IF a a2 A2 + a cl2jd2 ••• + «- a6 r 2, /// A , Ht T> , Ht a arAQ+a a2_o6... + a o-qA'q. But these impulsive wrenches give rise to an instantaneous twist velocity about a whence by § 80, we have, if A be a common factor, and a, b, c the principal radii of gyration + haß4 — a^j + + d4D4 + d3E, + a3F4, — haß2 = aJ^l2 + o..2B,2 + a3C.2 + a4D,2 + asE.2 + a6F.2, + hbß3 = a4As + a.2B3 + a3C3 + a4D3 + asEs + aeF3, — hbß4 = a4A4 + a.2B4 + a3C4 + a4D4 + a5E4 + a6#4) + = “1^5 + a2-®5 + “3^5 + a4^S + ar,^6 + a6-^5> — hcß3 = + a.2B3 + a3C3 + a4D3 + a3E6 + a6#6. Thus the linear relations are established. NOTE VIII. Remarks on § 268. It ought to have been mentioned that the relation between four points on a sphere used in this article is a well known theorem, see Salmon, Geometry of Three Dimensions, § 56 and Casey’s Spherical Trigonometry, § 111. It is also worth while to add that J'E1W is the function which on other grounds has been called the sine of the solid angle formed by the straight lines 1, 2, 3 (Casey, Spherical Trigonometry, § 28). The three formulæ of this article have been proved as they stand for sets of six co-reciprocal screws. Mr J. H. Grace has however kindly pointed out to me (1898) that the second of the formulæ would be also true for any set of five co-reciprocal screws, and the third would be true for any set of four co-reciprocal screws. We thus have for a set of four co-reciprocals Pi sin2 (234) + p.2 sin8 (341) + p3 sin3 (412) + p4 sin3 (123) = 0, where sin2 (234) is the square of the sine of the solid angle contained by the straight lines 2, 3, 4.