﻿the Duplex Har monograph. 501 



Epicyclic curves ma}- be drawn with a definite orientation 

 i£ we provide a method o£ fixing the initial positions of the 

 radius vectors o£ the circular motions. Thus the astroid 

 could be drawn with its cusps on the axes of coordinates 

 by providing B with two rods so as to release C and A 

 simultaneously, as was done in fig. 18, then similarly 

 releasing- 1). Counting; time from the instant when and A 

 are released, the equations to the trace would be 



x = a cos \_pt + 77 ] + a/3 cos [3pt + 7r] , 



y = a cos [p(t + 7r/2/>) + 7r] + aj'd cos [3p (t — irjQp) + it] . 



or x 2 l*+y 2 i z = (±a\S) 2 l\ 



Since the evolute of an ellipse may be derived from the 

 astroid by homogeneous strain, to draw the evolute of the 

 specified ellipse 



x 2 la 1 2 +f/b ] 2 =l, 



we must make the amplitudes of A and C equal to 



3(ai 2 — 6 1 2 )/4a 1 and {a\—b 2 )l^a l respectively, 



and those of B and D equal to 



3(a t 2 — 6i 2 )/l^i and {a^ — &i 2 )/!&i respectively, 



while the method of release and the frequencies of vibration 

 are unchanged. 



Figs. 23 and 24 are drawn with the amplitudes equal, 

 the ratio of the frequencies of the pairs of pendulums being 

 1 to 2 ; fig. 23 is the direct and 24 the retrograde epicyclic. 

 The equations to fig. 23, in which the circles are both anti- 

 clockwise, are 



x = a cos [p(t + 7r/2p)+ 6]+ a cos [2p(£ + tt/4jo) + 0], 



y = a cos [pt + 6] + a cos [2pt + (/>] , 



This curve is the trisectrix, the epitrochoid traced by a point 

 distant a from the centre of a circle of radius a/2 rolling on 

 a circle of equal radius. The equations to fig. 24 are 



x—a cos [pt + 0] +a cos [2p(t + 7r/4/>) + <£], 



^ = acos [p(t + 7r/ 2 p) + Q]+ a cos [2pt + <£]. 



This hypotrochoid is described by a point distant a from the 

 centre of a circle of radius a/2 rolling inside another circle 

 of radius 3a/2. 



Figs. 25 and 26 are related to figs. 23 and 24 like figs. 2L 

 and 22 are related to figs. 19 and 20. In fig. 25 the ampli- 

 tudes of the motions of A and B are twice those of C and D, 



