A Treatise on the Theory of Screws

290] DEVELOPMENTS OF THE DYNAMICAL THEORY. 307 Let a, ß, 7 be three screws of one system, and let t], %, % be the three corresponding screws, and, as usual, let represent the virtual coefficient of a and £. Then whenever the homography is chiastic:— ^a^ß^yr) = a^ß^yf- This is geometrically demonstrated when the following theorem is proved:— , If six points be inscribed on a circle, then the continued product of the three perpendiculars let fall from any point in a Pascal line formed from these six points upon three alternate sides of the corresponding hexagon is equal to the continued product of the three perpendiculars let fall from the same point on the other three sides. Let aa', ßß', yy' be the three pairs of sides, and write the equation aßy = o-'ß'y, then this represents a cubic curve through the nine points aa', a/3', and this cubic can only be the circle and the Pascal line. 289. Construction of Chiastic Homography on the Cylindroid. It is first obvious that, if two corresponding pairs of screws be arbitrarily selected, it will always be possible to devise one chiastic homography of which those two pairs are corresponding members. The circular construction shows this at once for, join AB' and A'B, they intersect at T, then the line TO is the homographic axis, and the correspondent to X is found by drawing A'X, and then AX' through the intersection of A'X and OT. 290. Homographic Systems on Two Cylindroids. The fundamental theorem for the two cylindroids is thus expressed:— Take any two screws, a and ß, on one cylindroid, and any two screws, r/ and on the other, it will then be possible to inscribe one, and in general only one chiastic homography on the two surfaces, such that a and y shall be correspondents, and also ß and For, write the general equation '®'a('&ß-il'&yl — ^aC^ßi^y^- If, then, a, ß, t), are known, and if y be chosen arbitrarily on the first cylindroid, it will then be always possible to find one, but only one, screw £ on the second cylindroid which satisfies the required condition. If a body had two degrees of freedom expressed by a cylindroid A, and if an arbitrary cylindroid B were taken, then an impulsive wrench ad- ministered by any screw on B would make the body commence to twist 20—2