A Treatise on the Theory of Screws

222 THE THEORY OF SCREWS. [214 add the products to the former equation. We can then equate the co- efficients of o0lt ...806 severally to zero, thus obtaining c/J? 2j>i pe + X2I1 + /iBi = 0, ■ Pq + X_4S + /z-B6 — 0. Choose next from the four-system any screw whatever of which the co- ordinates are </>!,... <£6. Multiply the first of the above six equations by </>1; the second by </>2, &c. and add the six products. The coefficients of X and /z vanish, and we obtain 9 (a. . 1 a A The coefficient of pe is however merely double the cosine of the angle between 0 and <f>. This is obvious by employing canonical co-reciprocals in which R = (01 + 0^ + (03 + 0f + (0S + 06f, whence ÆR dR d0i'" + ^d06 = 2 (</>! + </>2) (0j 4- 02) + 2 (</>3 + </>4) (03 4- 0,) + (^>5 4- <£(.) (0S 4- #6) = 2 cos We thus obtain the following theorem, which must obviously be true for other values of n besides four. If be any screw of an n-system and if 0 be a screw of stationary pitch in the same system then = cos (0</>) pe. Suppose that there were two screws of stationary pitch 0 and $> in an n- system. Then OTrø=COS (0^>)pe, OT^ = cos (Øø)#». If pe and are different these equations require that = 0; cos (0<f>) = 0; i.e. the screws are both reciprocal and rectangular and ‘ must therefore intersect. We have thus shown that if there are two stationary screws of different pitches in any n-system, then these screws must intersect at right angles. In general we learn that if any screw r/> of an «-system has a pitch equal to that of a screw 0 of stationary pitch in the same system, then 0 and must intersect. For the general condition = cos (0</>) pe