A Treatise on the Theory of Screws

144] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 131 It also follows that coQA' is constant; whence we have the following theorem :— Draw through the impulsive screw A a ray AH parallel to the homographic axis, then the ray from H to a fixed point fl on the homographic axis will cut the circle in the instantaneous screw A', and the acquired twist velocity will be inversely proportional to flzl'. If the twist velocity to be acquired by A’ from a unit impulsive wrench on A be assigned, then DA' is determined: there will be two screws A', and two corresponding impulsive screws, either of which will solve the problem. The diameter through D indicates the two screws about which the body will acquire the greatest and the least velocities respectively with a given intensity for the impulsive wrench. 143. Twist Velocities on the Principal Screws. The quantities a and ß, which are the twist velocities acquired by unit impulsive wrenches on the principal screws, can be expressed geometrically as follows (Fig. 22):— Let co be the twist velocity acquired on A' by the wrench on A, then, by the last article, aAY = a>A.'Y, ßAX=<oA'X; n AY AY “ : ß " AX : AX ■ whence This ratio is the anharmonic ratio of the four points X, Y, A, A', that is, of X, Y, 0, O'; whence, finally, a : ß " O’Y OY O'X ' OX ’ 144. Another Investigation of the Twist Velocity acquired by an Impulse. We have just seen that aAY=a>A'Y, ßAX = coA'X-, whence aß AX . AY = arA'X. A' Y. Let fall perpendiculars AP, AP’, HQ on the homographic axis (Fig. 24). Then, by the properties of the circle, AX. AY : AX. AY :: AP : AP'- so that aßAP = co1 AP'. 9—2