A Treatise on the Theory of Screws

333] VARIOUS EXERCISES. 361 If the corresponding impulsive screws are reciprocal, then 0 = X'X'^p^2 + pp’lp^ßy2 + vVl/pM + (X>" + X"/) + (XV' + X'V) + p'/z") S^V/31711 and two similar equations. Take the two conics whose equations are 0 = pax2 + p?y2 +pyz2 + 2tzapa>y + Z-^yXZ + 2^ßyyz, 0 = a^X^i3«!2 + y2^.p^ß2 + z^p^y2 + 2xytp13a1ß1 + 2æzXp1sa.1y1 + these conics will generally have a single common conjugate triangle. If the co-ordinates of the vertices of this triangle be X', p!, v ; X", p", v"; X"', p"', v"; then the equations just given in these quantities will be satisfied; and as there is only one such triangle, the required theorem has been proved. It can be easily proved that a similar theorem holds good for a pair of impulsive and instantaneous cylindroids. 333. Impulsive and Instantaneous Cylindroids. If a given cylindroid U be the locus of the screws about which a free rigid body would commence to twist if it had received an impulsive wrench about any screw on another given cylindroid V, it is required to calculate so far as practicable the co-ordinates of the rigid body. Let us take our canonical screws of reference so that the two principal screws of the cylindroid U have as co-ordinates 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0. The co-ordinates of any other screw on U will be «i, 0, as, 0, 0, 0. The cylindroid V will be determined by four linear equations in the co-ordinates of y. These equations may with perfect generality be written thus (§ 77), Vs = -4’71 + By2; V-i = A'yy + B'y2; y5 + A"yA + B"y2; y6 = A"'y1 + B'"y2, where A, B, A', B', A", B", A"', B'" are equivalent to the eight co-ordinates defining the cylin droid V. The screw on Ü with co-ordinates 1, 0, 0, 0, 0, 0