THE THEOEY OF SCKEWS. 



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 

 of movements by which the body can be transferred from one of these 

 positions to the other. It has 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 x l , ae z , x 3 , x 4&amp;gt; then the quadriplanar coordinates 

 2/i. 2/2&amp;gt; 2/s- 2/4 f 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 



y, = (11) x, + (12) x z + (13) x. + (14) x,, 

 y* = (21) x, + (22) x, + (23) x. A + (24) x t , 

 y 9 = (31) x, + (32) x, + (33) x a + (34) a? 4 , 

 y 4 = (41) ^ + (42) x z + (43) x, + (44) x.. 

 B. 1 



