A Treatise on the Theory of Screws

273] EMANANTS AND PITCH INVARIANTS. 287 If we substitute these values for alt ... an in the expression pa=tp1a^ we obtain the equation 0=p.2 cos2 al cos2 a2 ------ . Pi P* cos2 a6 P« . + 2po cos al ( pT cos al — dal sin al) Pi cos a6 (pe cos a6 — daS sin a6) + P« - 2 + ( p1 cos a 1 — dal sin al)2 ( p6cos a&-dals sin a6)2 Pi P« As al, &c., daj, &c., p{, &c. are independent of pa we must have the three co-efficients of this quadratic in pa severally equal to zero. 272. Three Pitches Positive and Three Negative. The equation cos2 al cos2 a2 cos2 a6 ------■+-------+••. + •-----=0 Pl Pl Ps also shows that three pitches of a set of six co-reciprocals must be positive and three must be negative. For, suppose that the pitches of four of the co-reciprocals had the same sign, and let a be a screw perpendicular to the two remaining co-reciprocals, then the identity just written would reduce to the sum of four positive terms equal to zero. From this formula and also 1 1 —I---- Pi Pi 1 P<i we have sin2 ai Pi sin2 a2 sin2 a 6 + ... +----------= 0. Pi ps 273. Linear Pitch Invariant Functions. We propose to investigate the linear functions of the six co-ordinates of a screw which possess the property that they remain unaltered notwith- standing an alteration in the pitch of the screw which the co-ordinates denote. It will first be convenient to demonstrate a general theorem which introduces a property of the six screws of a co-reciprocal system. The virtual coefficient of two screws is, as we know, represented by half the expression (pa + Pß) cos d — d sin 6, where pa and pß are the pitches, Ö is the angle between the two screws, and d the shortest perpendicular distance. The pitches only enter into