A Treatise on the Theory of Screws

198 THE THEORY OF SCREWS. (198— The angle through which the body has been rotated is (/2 + <72 + A2)i. and the distance of translation is af+ bg + ch (f2+g2 + W while the pitch of the screw is af+ bg + ch f2 + g'2 + h2 ’ Every distinct set of three quantities, (91; 0.,, 03, will correspond to a definite position of the rigid body, and to a group of such sets there will be a coiresponding group of positions. Let p denote a variable parameter, and let us consider the variations of the set, p0i, p02, p03, according as p varies. To each value of p a corresponding position of the rigid body is appropriate, and we thus have the change of p associated with a definite progress of the body through a series of positions. We can give geometrical precision to a description of this movement. The equations to the axis of the screw, as well as the expression of its pitch, only involve the ratios of a, b, c,f, g, h. We have also seen that these quantities are each linear and homogeneous functions of 0lt 0.3, 03. If, therefore, we substitute for 0lt 0.2> 03 the more general values P0i> p02, p03, the screw would remain unaltered, both in position and in pitch, though the angle of rotation and the distance of translation will each contain p as a factor. Thus we demonstrate that the several positions denoted by the set p0lt p02, p03 are all occupied in succession as we twist the body continuously around one particular screw. 199. The Plane Representation. All possible positions oi the body correspond to the triply infinite triad 0i, 02, 0S. If, for the moment, we regard these three quantities as the co-ordinates of a point in space, then every point of space will be correlated to a position of the rigid body. We shall now sort out the triply infinite multitude of positions into a doubly infinite number of sets each containing a singly infinite number of positions.