A Treatise on the Theory of Screws

270] EMANANTS AND PITCH INVARIANTS. 285 269. Property of the Pitches of n Co-reciprocals. The theorem just proved can be extended to show that the sum of the reciprocals of the pitches of n co-reciprocal screws, selected from a screw system of the nth order, is a constant for each screw system. Let A be the given screw system, and B the reciprocal screw system. Take 6 — n co-reciprocal screws on B, and any n co-reciprocal screws on A. The sum of the reciprocals of the pitches of these six screws must be always zero; but the screws on B may be constant, while those on A are changed, whence the sum of the reciprocals of the pitches of the n co-reciprocal screws on A must be constant. Thus, as we have already seen from geometrical considerations, that the sum of the reciprocals of the pitches of co-reciprocals is constant for the screw system of the second and third order (§§ 40, 176), so now we see that the same must be likewise true for the fourth, fifth, and sixth orders. The actual value of this constant for any given screw system is evidently a characteristic feature of that screw system. 270. Theorem as to Signs. If in one set of co-reciprocal screws of an n-system there be k screws with negative pitch and n — k screws with positive pitch, then in every set of co-reciprocal screws of the same system there will also be k screws with negative pitch and n — k screws with positive pitch. To prove this we may take the case of a five-system, and suppose that of five co-reciprocals Alt A2, t!3, _44j As the pitches of three are positive, say nii, mf, mf, while the pitches of the two others are negative, say — ml, - mf Let >S be any screw of the system, then if 0lt ... 0S be its co-ordinates with respect to the five co-reciprocals just considered, we have for the pitch of 0 the expression ml 01 + mlØr + m320I — ml 01 — mfØ?. Let us now take another set of five co-reciprocals Blt B.it B3, Bit Bs belonging to the same system, then the pitches of three of these screws must be + and the pitches of two must be —. For suppose this was not so, but that the five pitches were, let us saynx2, n2a, ws3, nt> — n3. Let the co-ordinates of S with respect to these new screws of reference be </>1( </>2,... </>6, then the pitch will be nityl + n2a</>22 + nffå + nif,2 - n^2.