I4 2 KINEMATICS. [263. 



complete quadrangle, or four-point, C^f^C^C^ nas s ' lx sides, 

 viz. the six lines C 41 C lzt C\>y& ^23^34' ^34^41' ^12^34' ^23^41 

 joining its four vertices ; and these six sides of the quadrangle 

 pass through the six vertices of the quadrilateral, respectively. 



It remains to prove that C 1Z C 8 passes through 5 and that C^C^ 

 passes through 6. 



Now the velocity of 2 can be expressed by Wj C 12 2 and also 

 bya} 2 -C^2\ hence C l2 2/C 2B 2 = a) 2 /(0 1 ; similarly C^/C^ = 

 ft> 3 /ft> 2 . We have therefore, by the proposition of Menelaus,* for 

 the intersection of 2 3 with C 12 C M : 



5C\ 2 = ft)g 



5 Q 4 w i 



The same value is obtained by determining the intersection of 

 i 4 with C 12 C 34: ; the two intersections must therefore coincide. 

 The proof for the point 6 is analogous. 



* If the sides of a triangle ABC be cut by any transversal, in the points A', B', C', 



p> Al /""/?' A C 1 



then =~- ~~- =-i. See J. CASEY, Sequel to Euclid, London, Longmans, 

 A C Jj A C JD 



1882, p. 69. 



