A Treatise on the Theory of Screws

145] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 133 and thus obtain the construction used in § 142. A similar construction is obtained when fl' is at infinity. The two points, A and A', will divide the arc cut off by XY in a constant anharmonic ratio, for the pencil H (XOAl'Y) always preserves the same anharmonic ratio as H moves round the circle. 145. A Special Case. If ?? be an impulsive screw, and if a be the corresponding instantaneous screw, it will not usually happen that when a is the impulsive screw y is the corresponding instantaneous screw. If, however, in even a single case, it be true that the impulsive screw and the instantaneous screw are interchange- able, then the relation will be universally true. Let fl and ft' (Fig. 26) be a pair of points belonging to the system described in § 144. Then A being given, A' is found. If A' is similarly to Fig. 26. determine A, then the figure shows that 11 must lie on the polar of fl', and, consequently, fl and fl' are conjugate points with respect to the circle; or, what comes to the same thing, they divide XY harmonically. The same must be true of each pair of points 12 and H', and therefore of 0 and O', and we have the following theorem :— If the points 0 and O' be harmonic conjugates of the points where the homo- graphic axis intersects the circle, then every pair of instantaneous and impulsive screws on the cylindroid are interchangeable. We might, perhaps, speak of this condition of the system as one of dynamical involution. In this remarkable case an impulsive wrench of unit intensity applied to one of the principal screws of inertia will generate a velocity equal and opposite to that which would have been produced if the wrench had been applied to the other principal screw. The construction