A Treatise on the Theory of Screws

243-246] HOMOGRAPHIC SCREW SYSTEMS. 263 There are in general three points, which coincide with their corre- spondents. These are found by putting Ä = P«i; ß3 = p^', ß3 = pa3. Introducing these values, and eliminating a1, a2, as, we obtain the following equation for p :— 0= (11) -p, (12), (13) I (21), (22) -p, (23) . (31), (32), (33) -p I If we choose these three points of the vertices of the triangle of reference, the equations relating y with x assume the simple form, &=/2a2; /33=/3a3, where fufnf;, are three new constants. 245. Homographic Screw Systems. Given one screw a, it is easy to conceive that another screw ß correspond- ing thereto shall be also determined. We may, for example, suppose that the co-ordinates of ß (§ 34) shall be given functions of those of a. We might imagine a geometrical construction by the aid of fixed lines or curves by which, when an a is given, the corresponding ß shall be forthwith known : again, we may imagine a connexion involving dynamical conceptions such as that, when a is the seat of an impulsive wrench, ß is the instantaneous screw about which the body begins to twist. As a moves about, so the corresponding screw ß will change its position and thus two corresponding screw systems are generated. Regarding the connexion between the two systems from the analytical point of view, the co-ordinates of a and ß will be connected by certain equations. If it be invariably true that a single screw ß corresponds to a single screw a, and that conversely a single screw a corresponds to a single screw ß; then the two systems of screws are said to be homographic. A screw a in the first system has one corresponding screw ß in the second system; so also to ß in the first system corresponds one screw a' in the second system. It will generally be impossible for a and a to coincide, but cases may arise in which they do coincide, and these will be discussed further on. 246. Relations among the Co-ordinates. From the fundamental property of two homographic screw systems the co-ordinates of ß must be expressed by six equations of the type— ••• ’s) Ä =/<;(«!, ... a6).