A Treatise on the Theory of Screws

THE THEORY OF SCREWS. INTRODUCTION. The Theory of Screws is founded upon two celebrated theorems. One relates to the displacement of a rigid body. The other relates to the forces which act on a rigid body. Various proofs of these theorems are well known to the mathematical student. The following method of considering them may be found a suitable introduction to the present volume. ON THE REDUCTION OF THE DISPLACEMENT OF A RIGID BODY TO ITS SIMPLEST FORM. Two positions of a rigid body being given, there is an infinite variety °f movements by which the body can be transferred from one of these positions to the other. It lias been discovered by Chasles that among these movements there is one of unparalleled simplicity. He has shown that a free rigid body can be moved from any one specified position to any other specified position by a movement consisting of a rotation around a straight line accompanied by a translation parallel to the straight line. Regarding the rigid body as an aggregation of points its change of place amounts to a transference of each point P to a new point Q. The initial and the final positions of the body being given each point P corre- sponds to one Q, and each Q to one P. If the coordinates of P be given then those of Q will be determined, and vice versa. If we represent P by its quadriplanar coordinates xlt x2, xa, x4, then the quadriplanar coordinates 36, ^2, 2/3, y4 of Q must be uniquely determined. There must, therefore, be equations connecting these coordinates, and as the correspondence is essentially of the one-to-one type these equations must be linear. We shall, therefore, write them in the form y4 = (11) Xj + (12) x2 + (13) x3 4- (14) x4, 3/2 = (21) x, +(22) x.j+ (23) x3 + (24) x4, y3 = (31) x4 + (32) x2 + (33) x3 + (34) x4, yt = (41) x4 + (42) x2 + (43) x3 + (44) x4. B. 1