A Treatise on the Theory of Screws

136] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 125 but, from the property of the circle, BY. XO' = AX. BO'; A Y. XO' = BX. AO'-, whence AB-.XY. XO' = ?LY2. A B. BO' + BX2 .AB. AO', from which we obtain, as before, 0-«/ = d3wa2 + ß"itß2. 136. Impulsive Screws and Instantaneous Screws. A rigid body having two degrees of freedom lies initially at rest. It is suddenly acted upon by an impulsive wrench of large intensity acting for a short time. The body will, in,general, commence to move by twisting about some screw on the cylindroid, and the kinetic problem now to be studied is the following:—Given the impulsive screw, and the intensity of the impul- sive wrench, find the instantaneous screw and the acquired twist velocity. The problem will be rendered more concise by the conception of the reduced wrench (§ 96). It is to be remembered, that as the body is only partially free, there are an infinite number of screws on which wrenches would make the body commence to twist about a given screw on the cylin- droid. For, let 0 be an impulsive screw situated anywhere, and let an impulsive wrench on 0 cause the body to commence to move by twisting about some screw, a, on the cylindroid. Let X, /t, v, p be any four screws reciprocal to the cylindroid. Then any wrench on a screw belonging to the system defined by these five screws will make the body commence to move by twisting about a. Let e be that one screw on the cylindroid which is reciprocal to 0, then e is reciprocal to the whole system defined by X, p., v, p, 0, and, conversely, each screw of this system will be reciprocal to e. VV e thus see that any screw, wherever situated, provided only that it is reciprocal to e, will be an impulsive screw corresponding to a as an instan- taneous screw. Any one of this system may, with perfect generality, be chosen as the impulsive screw. Among them there is one which has a special feature. It is that screw, </>, on the cylindroid which is reciprocal to p ; and hence we have the following theorem (§ 128): Given any screw, a, on the cylindroid, then there is in general another screw, </>, also on the cylindroid, such that an impulsive wrench administered on </> will make the body twist about a. This correspondence of the two systems of screws must be of the one-to- one type; for, suppose that two impulsive screws on the cylindroid had the same instantaneous screw, it would then be possible for two impulsive wrenches, of properly chosen intensities on two different screws, to produce