A Treatise on the Theory of Screws

 358 THE THEORY OF SCREWS. [329- As y belongs to the three-system V we must have the six co-ordinates of y connected by three linear equations (§ 77); solving these equations we have % = + By 3 + Gys, yi = A'y1 +B'ys + C'y6, ye = A'f + B"y, + C”ys. The nine coefficients A, B, C, A', B', G', A'', B", G" are essentially the co- ordinates of the three-system I7. We now seek the co-ordinates of the rigid body in terms of these quantities. Take the particular screw of U which has co-ordinates 1, 0, 0, 0, 0, 0. Then the co-ordinates of the corresponding impulsive screw are ylt y2< ... where - en.yl = p,2 — a2; + eby, = az, — bz, — ; + ecy, = — ay, + cy, — li ; - eay, = pi- a2; - ebVi = az, + bz, - Z2a; - ecy, = - ay, - cy, - Z22. Since by hypothesis this is to belong to V, the following equations must be satisfied: °? ~ P" = A a"±pi + B az» ~bz,-lj cy,- ay, -li a a b c ’ - az, - bz, + lj = A, a2 + pi + ß, az, - bz, -If cy, - ay, - I2 b a b c ’ aV<> + cy, + lj =A„ a2 + p,2 + ß„ az, - bz, - I2 c„ cy, - ay, - I2 cab c In like manner by taking successively for a the screws with co-ordinates 0, 0, 1, 0, 0, 0 and 0, 0, 0, 0, 1, 0, we obtain six more equations of a similar kind. As these equations are linear they give but a single system of co-ordinates x,, y„ z,, l2, I2, I2, pi, pi, pi for the rigid body. The theorem has thus been proved, for of course if three screws of U correspond to three screws of V then every screw in U must have its correspondent restricted to V. 330. Construction of Homographic Correspondents. If the screws in a certain three-system U be the instantaneous screws whose respective impulsive screws form the three-system V, then when three pairs of correspondents are known the determination of every other pair of correspondents may be conveniently effected as follows.