A Treatise on the Theory of Screws

31] SCREW CO-ORDINATES. 33 30. The Intensity of the Resultant may be expressed in terms of the intensities of its components on the six screws of reference. Let a be any screw of pitch pa, let p„ p2, Szc. p6 be the pitches of the six screws of reference &>1; co,, ... coe; then taking each of the screws of refer- ence in succession, for y in § 29, and remembering that the virtual coefficient of two coincident screws is simply equal to the pitch, we have the following equations:— = a/'pi + a2"OTi2 + ... + a6"’w16, a"wB8 = + ... + «6"wM4- ae"pe. But taking the screw p in place of y we have Ct pa Ot j ^"«6 • Substituting for wal ... wall from the former equations, we deduce paa"s = 2 (jpi«/'2) + 22 (ai"«2"w12). This result may recall, the well-known expression for the square of a force acting at a point in terms of its components along three axes passing through the point. This expression is of course greatly simplified when the three axes are rectangular, and we shall now show how by a special disposition °f the screws of reference, a corresponding simplification can be made in the formula just written. 31. Co-Reciprocal Screws. We have hitherto chosen the six screws of reference quite arbitrarily; we now proceed in a different manner. Take for co,, any screw; for w2, any screw reciprocal to cop, for <a3, any screw reciprocal to both wj and co.,; for <a4, any screw reciprocal to , <a2, <o3; for <»s, any screw reciprocal to Wj, co,, w3, w4; for <u6, the screw reciprocal to Wj, co,, co3, co4, cos. A set constructed in this way possesses the property that each pair of screws is reciprocal. Any set of screws not exceeding six, of which each pair is reciprocal, may be called for brevity a set of co-reciprocals*. Thirty constants determine a set of six screws. If the set be co- reciprocal, fifteen conditions must be fulfilled; we have, therefore, fifteen elements still disposable, so that we are always enabled to select a co- reciprocal set with special appropriateness to the problem under con- sideration. * Klein has discussed (Math. Ann. Band n. p. 204 (1869)) six linear complexes, of which eacli Pair is in involution. If the axes of these complexes be regarded as screws, of which the “ Hauptparameter” are the pitches, then these six screws will be co-reciprocal. B, 3