A Treatise on the Theory of Screws

299] DEVELOPMENTS OF THE DYNAMICAL THEORY. 319 X1 = 0, X3 = 0, Xs = 0. Combining these conditions with the last, we draw the general conclusion that ^=0, X2 = 0, Z3 = 0, Xt = 0, X5 = 0, X6 = 0; or pe<f>1 = a01; pe<l>2 = — a03', pøtys — bØs', pe<l>i = — b04 &c. Thus we demonstrate that if a pair of screws 0, <p satisfy the six conditions, they stand to each other in the relation of impulsive screws and instantaneous screws. 299. Cylindroid Reduced to a Plane. Suppose that the family of rigid bodies be found which make a, p and ß, £ impulsive and instantaneous. Let there be any third screw, y, and let us seek for the locus of its impulsive screw, for all the different rigid bodies of the family. £ must satisfy the four equations —cos (y?;) 4--------cos (“f) = cos (ai?) ' cos(yf) —vx cos (y£) + —cos (/3f) = cos(/3£) cos (7?) P«. n Py „ COS (arf) cos (7?) af’ Pß Py cos (ߣ) cos (7?) As there are four linear equations in the coordinates of we have the following theorem. If a, t) and ß, g be given pairs of impulsive and instantaneous screws, then the locus of £ the impulsive screw corresponding to y, as an instan- taneous screw, is a cylindroid. But this cylindroid is of a special type. It is indeed a plane surface rather than a cubic. The equations for f can have this form :— cos (af) = A cos (yf), = C cos (7^), cos (/?£) =-B cos (7£), = Deos (7$’), in which A, B, C, D are constants. The fact that cos (af) and cos (7^) have one fixed ratio, and cos (/3£) and cos (yt) another, shows that the direction of £ is fixed. The cylindroidal locus of £, therefore, degenerates to a system of parallel lines. At first it may seem surprising to find that is constant. But the