﻿A Kinematic Solution of Equations. 893 



Tins will result in a number of kite-shaped figures whose 

 diagonals joining equal angles are in the ratio R ... R«. 



Now suppose bars to take the place of the lines drawn, 

 not ending, however, where the kites join, but continuous 

 across such points from the tail-end of one kite to the head 

 of the next ; joints are supposed at all points where such 

 bars either join or cut one another, of the nature of pins 

 round which the bars can turn. 



Such a system is a linkage capable of a proper motion 

 which will always maintain the chords R ... R n in the same 

 ratio among themselves, as well as equiangularity in the 

 whole polygonal train, though the angle between any two 

 consecutive chords will vary. 



Let A and B be the extremities of the linkage. Take a 

 pair of rectangular axes Ox, Oy, and suppose A to be pinned 

 down at 0, and make the other extremity of R lie in Ox, 

 and let 6 be the angle between any two consecutive sides of 

 the polygon externally. 



Then clearly the projection of AB upon line Ox is pro- 

 portional to 



B, o + Ricos0 + Rocos20+ 4-R„cosn0. 



But the equation requires that this shall be zero. 



If, therefore, the proper motion is given to the linkage, 

 always keeping R in Ox, until the point B is found in 0//, 

 in that position R T makes with Ox the angle whose cosine 



