﻿<S06 Mr. T. H. Blakesley: A Kinematic 



The last term will appear as +No? n in the first bracket if 

 is even, in the second if n is odd. 

 Let 



, n sin 6 



x = tan = ^, 



cos 



and substitute for x this value. Then 

 sin 2 „ sin 4 



°={ A + C cos4 +E co^ 



L cos 6 cos rf cos 



Reducing each line to a common denominator, which will 

 be some power of cos 0, and cancelling out as far as possible, 

 it is to be noticed that if n is even there will remain cos in 

 the denominator of the first line; if n is odd, in that of the 

 second. Multiplying through by cos 0, and making the 

 proper substitutions to obtain cosines in the first line and 

 sines in the second, which will always be found possible, the 

 equation will take the forms : — 



If n is even, = { R + R 2 cos 20 + ... + R„ cos n0} 



-{r 2 sin20 + r 4 sin 40 + ... + r, l sin??#}. 



If n is odd, 0= {Rjcos + R 3 cos 30 + ... +R„ cos n0} 

 — {r x sin + r 3 sin 30 f- ... +r n smn0}. 



All the coefficients in these expressions are linear functions 

 of the coefficients in the original equation which are readily 

 determined. 



The first line in either case is the projection of a line 

 joining the first and last points of a linkage having 29 as 

 the external angle between the sides of the polygon, upon 

 the line Ox. 



The quantity within the bracket of the second line is the 

 projection of another linkage having the same external angle 

 20, in its polygon, upon the line Oy. 



The two trains may obviously be actuated at once from the 

 same first kite. 



The criterion derived from the equation is that the two 

 projections shall be equal. When this is the case in the 

 proper motion of the system, the root of the equation is tan 0. 

 being half the angle between the two homologous sides of 

 two consecutive kites. 



The line joining the feet of the perpendiculars of projection 

 must make an angle of —45° with Ox. 



