﻿502 Dr. J. H. Vincent and Mr. C. W. Jude on 



the two direct circular motions having radii inversely as 

 their frequencies. The equations to fig. 25 are 



x = a cos [p(t+ ir/2p) + 6] + a/2 cos [2p(t + tt/4/?) + (/>] , 

 y = a cos [pt -f 0] + a/2 cos [2pt + <£] . 



The origin of coordinates was marked by the needle on the 

 prepared plate with the pendulums at rest. This curve is the 

 cardioide, the epicycloid traced by a point on the circum- 

 ference of a circle of radius a/2 rolling on an equal circle. 



To draw fig. 26 one of the component circular motions of 

 fig. 25 must be reversed. The equations to the trace are 



x = a cos ipt + 0] -f a/2 cos [ 2p(t + ir/^p) + <£] , 



?/ = acos [p(t + 7r/2p) -f<9] + a/2 cos \2pt + (p]. 



The curve is the tri-cusp, the three-cusped hypocycloid 

 drawn by a point on the circumference of a circle of radius 

 a/2 rolling inside a circle of radius 'da/2. 



Frequencies slightly different. 



In the foregoing examples of epicyclics the frequencies 

 have been in simple ratio. If this ratio is slightly departed 

 from the whole trace may be regarded as a family of curves, 

 each member of which approximates to that proper to the 

 simple ratio, while each successive member is rotated about 

 the origin. The only examples of this character which we 

 shall give have the ratio of the frequencies nearly unity. 



In figs. 21 and 28 the frequencies of the first pair A and B 

 are slightly less than those of C and D. The amplitudes 

 are equal. The anti-clockwise circular motion due to C and 

 D has slightly greater angular velocity than the clockwise 

 motion. The equations to the trace are 



/6' = acos [pt + 0] +acos[{p + h}(t + 7r/2{p-\-h})-\-cj)] } 



j/ = acos [p{t + 7r/2p)+6'] +acos [{/> + 7i}* + 0], 



in which h is small compared with p. The trace lasts for a 

 time 2ir/h in fig. 27 and tt/Ii in fig. 28. The curve is the 

 hypotrochoid traced by a point distant a from the centre 

 of a circle of radius ap/{p J rh} rolling inside a fixed circle of 

 radius a{2p + h}/{p + h}. 



" This is intimately connected with the explanation of two 

 sets of important phenomena, — the rotation of the plane of 

 polarization of light, by quartz and certain fluids on the one 

 hand, and by transparent bodies under magnetic forces on 

 the other It will also appear in kinetics as the path 



