200] PLANE REPRESENTATION OF THE THIRD ORDER. 190 



If we fix our glance upon the screws about which the body is free to 

 twist, the principle of classification will be obvious. Take an arbitrary triad 



01, 2, 03, 



and then form the infinite group of triads 



for every value of p from zero up to any finite magnitude : all these triads 

 will correspond to the positions attainable by twisting about a single screw. 

 We may therefore regard 



ly 0. 2 , 6 3 



as the co-ordinates of a screw, it being understood that only the ratios of 

 these quantities are significant. 



We are already familiar with a set of three quantities of this nature 

 in the well-known trilinear co-ordinates of a point in a plane. We thus 

 see that the several screws about which a body with three degrees of 

 freedom can be twisted correspond, severally, with the points of a plane. 

 Each of the points in a plane corresponds to a perfectly distinct screw, 

 about which it is possible for a body with three degrees of freedom to be 

 twisted. Accordingly we have, as the result of the foregoing discussion, the 

 statement that 



To each screw of a three-system corresponds one point in the plane. 

 To develope this correspondence is the object of the present Chapter. 



200. The Cylindroid. 



A twist of amplitude 6 on the screw 6 has for components on the three 

 screws of reference 



0i, 0. 2 , S ; 



a twist of amplitude &amp;lt; on some other screw &amp;lt; has the components 



When these two twists are compounded they will unite into a single twist 

 upon a screw of which the co-ordinates are proportional to 



If the ratio of to & be X, we see that the twists about and &amp;lt; unite into 

 a twist about the screw whose co-ordinates are proportional to 



01 + \&amp;lt;f&amp;gt; ly 6. 2 + X._,, 0-j + \&amp;lt;&amp;gt; s . 



By the principles of trilinear co-ordinates this point lies on the straight line 

 joining the points and &amp;lt;. As the ratio \ varies, the corresponding screw 



