A Treatise on the Theory of Screws

266] EMANANTS AND PITCH INVARIANTS. 279 This can be verified in the following manner. We have K=pX'3, d\j d Xj d \ 2 and, therefore, if X be reciprocal to the first screw of reference, the formula to be proved is d Xj d X; A few words will be necessary on the geometrical signification of the differentiation involved. Suppose a Dyname X be. referred to six co-ordinate screws of absolute generality, and let us suppose that one of these co- ordinates, for instance X1( be permitted to vary, the corresponding situation of X also changes, and considering each one of the co-ordinates in succession; we thus have six routes established along which X will travel in correspond- ence with the growth of the. appropriate co-ordinate. Each route is, of course, a ruled surface; but the conception of a surface is not alone adequate to express the route. We must also associate a linear magnitude with each generator of the surface, which is to denote the pitch of the corresponding screw. Taking X and another screw on one of the routes, we can draw a cylindroid through these two screws. It will now be proved that this cylindroid is itself the locus in which X moves, when the co-ordinate cor- related thereto changes its value. Let 0 be the screw arising from an increase in the co-ordinate Xjj a wrench on 0 of intensity 0' has components of intensities 0/,)... 0,'' ■ A wrench on X has components X/,... X8 . But from the nature of the case, V V” V If therefore 0" be suitably chosen, we can make each of these ratios — 1, so that when 0" and X" are each resolved along the six screws of reference, all the components except 0”, — X/' shall neutralize. But this can only be possible if the first reference screw lie on the cylindroid containing 0 and X. Hence we deduce the result that each of the six cylindroids must pass through the corresponding screw of reference; and thus we have a complete view of the route travelled by a screw in correspondence with the variation, of one of its co-ordinates. Let the six screws of reference be 1, 2, 3, 4, 5, 6. Form the cylindroid (X, 1), and find that one screw on this cylindroid which has with 2, 3,4, 5, 6, a common reciprocal (§ 26). From a point 0 draw a pencil of four rays parallel to four screws on the cylindroid. Let OA be parallel to one of the principal screws; OX be parallel to X, Oq to y, and Oh to the first screw of reference.