A Treatise on the Theory of Screws

63] THE KEPRESENTATION OF THE CYLINDROID BY A CIRCLE. 57 The negative sign has no significance for our present purpose, and hence we have the following theorem : The virtual coefficient of two screws is equal to the cosine of the angle subtended by their chord, multiplied into the perpendicular from the pole of the chord on the axis of pitch. This is, perhaps, the most concise geometrical expression for the virtual coefficient. It vanishes if the perpendicular becomes zero, for then the chord must pass through the pole of the pitch axis, and the two screws be reciprocal. The cosine enters the expression in order that its evanescence, when C = 90°, may provide for the circumstance that the perpendicular is then infinite. This result is easily shown to be equivalent to that of the last article by the well-known theorem :— If any two chords be drawn in a circle, then the cosine of the angle sub- tended by the first chord, multiplied into the perpendicular distance from its pole to the second chord, is equal to the cosine of the angle subtended by the second chord, multiplied into the perpendicular from its pole to the first chord. It follows that the virtual coefficient must be equal to the perpendicular from the pole of the axis of pitch upon the chord joining the two screws, multiplied into the cosine of the angle in the arc cut off by the axis of pitch. This is the expression of § 61, namely, OG 63. Application of Screw Co-ordinates. It will be useful to show how the geometrical form for the virtual coefficient is derived from the theory of screw co-ordinates. Let alt a„, and ßlt ß., be the co-ordinates of two screws on the cylindroid; then, if the screws of reference be reciprocal, the virtual coefficient is (§ 37) Pi«ißi + p2a2ß2. Let A, B (Fig. 14) be the screws of reference, and let C and C' be the two screws of which the virtual coefficient is required. Let PQ be the axis of pitch of which 0 is the pole, then 0 lies on AB, as the two screws of reference are reciprocal (§ 58). As AB is divided harmonically at 0 and H, we have AO : OB :: HA : HB :: AP : BQ :: Pl ; p2- whence 0 is the centre of gravity of masses — , ~ at A and B, respectively.