138] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 127 



those components must, when compounded, produce the twist velocity &&amp;gt; about 

 X , and, accordingly, we have 



. BX . B X AX . A X 



Retaining A, B, A , B , as before, let us now introduce a second pair of points, 

 Y and F , instead of X and X , and writing &&amp;gt; instead of o&amp;gt;, we have 



.BY .,FT 



_ 

 a AB ~ * A B AB ~ A B 



whence, eliminating a, /3, &&amp;gt;, o&amp;gt; , we have 



5Z BY B X FT 

 AX : AY :: A X : A Y&quot; 



As the length of a chord is proportional to the sine of the subtended 

 angle, we see that the anharmonic ratio of the pencil, subtended by the four 

 points A, B, X, Fat a point on the circumference, is equal to that subtended 

 by their four correspondents, A , B , X , Y . We thus learn the following 

 important theorem : 



A system of points on the representative circle, regarded as impulsive 

 screws, and the corresponding system of instantaneous screws, form two homo- 

 graphic systems. 



138. The Homographic Axis. 



Let A, B, C, D (fig. 21) represent four impulsive screws, and let A , B , 

 C , D be the four corresponding instantaneous screws. Then, by the well- 



Fig. 21. 



known homographic properties of the circle, the three points, L, M, N, will 

 be collinear, and we have the following theorem : 



