A Treatise on the Theory of Screws

278] EMANANTS AND PITCH INVARIANTS. 295 but we know that the last term is itself a pitch invariant, and hence we have the following result. If pa be the pitch of a screw a expressed in terms of the co-ordinates, and if Ra denote the function a/+ ... + a82 + 2ajOj cos (12) 4-... = 1, then the several functions d / \ d / p^ \ da, vt/nj ’ da6 \VrJ remain unaltered if instead of a,... a,. the co-ordinates of any other screw on the same straight line as a should be substituted. This is easily verified by the known formulæ that if a be any screw and 3 another screw on the same axis as a whose pitch is pa 4- x, then „ x dR a x dR da, 4p6 aaG whence — (= 2p0 -pe cos (01) = 2piai + - . - (?« + x) cos (al) deAjRf 2 da, , x d / p„. \