382 THE THEORY OF SCREWS. [348, 



It thus appears that four straight lines in one system may be chosen 

 arbitrarily to correspond respectively with four straight lines in the other 

 system ; and that one force being chosen on one of these straight lines in 

 one system, the corresponding force may be chosen arbitrarily on the corre 

 sponding straight line in the other system. This having been done, the 

 relation between the two systems is completely defined. 



From the case of parallel projections in two planes it is easy to pass to 

 the case which will serve our present purpose. Instead of the straight lines 

 in the two planes we shall take screws in two ^-systems. Instead of the 

 forces on the lines we may take either twists or wrenches on the screws. 

 More generally it will be better to use Plucker s word &quot; Dyname,&quot; which we 

 have previously had occasion to employ ( 260) in the sense either of a twist 

 or a wrench, or even a twist velocity. We shall thus have a Dyname in one 

 system corresponding to a Dyname in the other. 



Let us suppose that a Dyname on a screw of one n-system corresponds 

 uniquely to a different Dyname on a screw of another ?i-system. The two 

 n-systems may be coincident but we shall treat of the general case. 



In the first place it can be shown that if any number of Dynames in 

 the first system neutralize, their corresponding Dynames in the second 

 system must also neutralize. Take n screws of reference in one system, 

 and also n screws of reference in the corresponding system. Let 6 be the 

 Dyname in one system which corresponds to (f&amp;gt; in the other ; 6 can be 

 completely resolved into component Dynames of intensities 1} ... 6 n on the 

 n screws of reference in the first system and in like manner &amp;lt;f&amp;gt; can be resolved 

 into n components of intensities fa, ... &amp;lt;f&amp;gt; n on the screws of reference in the 

 second system (n = &amp;lt; 6). 



From the fact that the relation between and &amp;lt;f&amp;gt; is of the one-to-one 

 type the several components &amp;lt;j&amp;gt; l} ...&amp;lt;f&amp;gt; n are derived from 1 ,...0 n by n equations 

 which may be written 



fa t = ( ? *1) 0, + ( W 2) ,... S 4 (nn) 6 n , 



in which (11), (12), &c. must be independent of both and &amp;lt;, for otherwise 

 the correspondence would not be unique. 



If there be a number of Dynames in the first system the sums of the 

 intensities of their components on the n screws of reference may be expressed 

 as S#!, ... ^0 n respectively. In like manner the sums of the intensities of 

 the components of their correspondents on the screws of reference of the 

 second system may be represented by &quot;fa, ... 2&amp;lt;/&amp;gt; n respectively. We therefore 



