198 THE THEORY OF SCREWS. [198- 



The angle through which the body has been rotated is 



(/ 2 + &amp;lt;7 2 + /* 2 )&quot;, 

 and the distance of translation is 



af+ bg + ch 



while the pitch of the screw is 



af+bg + ch 



Every distinct set of three quantities, d l , #,, 3 , will correspond to a 

 definite position of the rigid body, and to a group of such sets there will be 

 a corresponding group of positions. Let p denote a variable parameter, and 

 let us consider the variations of the set, 



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 l , #,, 3 . If, therefore, we substitute 

 for #j, # 2 , # 3 the more general values 



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 p6 l} 

 p0. 2 , pO s are all occupied in succession as we twist the body continuously 

 around one particular screw. 



199. The Plane Representation. 



All possible positions of the body correspond to the triply infinite triad 



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. 



