A Treatise on the Theory of Screws

295] DEVELOPMENTS OF THE DYNAMICA.L THEORY. 313 and of X' aj cos O' + /3j sin O', ... a6cos O' + ß6sin Ö'. In like manner, let p and p' be two screws on a cylindroid, of which the two principal screws are p and Let </>, <f>' be the angles which p and p' make respectively with £. Then the co-ordinates of p are cos </> + sin . r/a cos </> + & sin (f>, and of p cos </>' + £ sin ye cos </>' + £6 sin </>', &c. We shall now suppose that the two cylindroids a, ß and p, % are so circumstanced that the latter is the locus of the impulsive wrenches cor- responding to the several instantaneous screws on the former with respect to the rigid body which is to be regarded as absolutely free. We shall further assume that p is the impulsive screw which has X as its instantaneous screw, and that the relation of p' to X' is of the same nature. If, however, the four screws X, X.', p, p' possess the relations thus indi- cated, it is necessary that they satisfy the conditions already proved (§ 281). These are two-fold, and they are expressed by the following equations, as already shown: — —cos (Xp) + —7*', , cos (Xp') = 2-ctååS cos (Xp) cos (X p ) Pk _ Pk cos (Xp) p cos (X'p) ’ We shall arbitrarily choose X' and p, so as to satisfy the conditions CT A'p = 0, W xp' = 0, and thus the second of the two equations is satisfied. These two equations will give O' as a function of </>, and </>' as a function of 0. We can thus eliminate 0' and </>' from the first of the two equations, and the result will be a relation connecting 0 and <f>. This equation will exhibit the relation between any instantaneous screw 0 on one cylindroid, and the corresponding impulsive screw </> on the other. It will be observed that when the two cylindroids are given, the required equation is completely defined. The homographic relations of p and X is thus completely determined by the geometrical relations of the two cylin- droids.