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



A and B. The circular motions are both clockwise. The 

 trace is the epitrochoid drawn by a point at a distance a 

 from the centre of a circle of radius a/3 rolling on a fixed 

 circle of radius 2a/3. In drawing such curves, one pair of 

 pendulums having been suitably released, the other pair was 

 started at a time most convenient to the experimenter, so 

 that the orientation of the diagram in such figures is not 

 determinate. Further, the commencement of the drawing- 

 was also arbitrary since we put the pen down so as to prevent 

 the overlap interfering with an interesting portion of the 

 trace. If, then, we insert and <fi in the corresponding- 

 pairs of terms, the equations to the trace may be written 



a = acos [pt + 0] + a cos \_2>pt + </>], 



y = acos [p(t + 7r/2p) + 0] + a cos [3p(t + 7r/6p) +<£]. 



In figs. 19 to 26 the duration of the trace is the period of 

 the slower pair of pendulums. 



All else being unchanged, the circle drawn by C and D 

 is reversed in fig. 20. The trace is the hypotrochoid de- 

 scribed by a point distant a from the centre of a circle of 

 radius a/3 which rolls inside a fixed circle of radius 4a/3. 

 The equations to the trace are 



x= a cos [p>t + 62 +acos [3p(t + 7r/6p) + (f)\ 



y = a cos \_p(t + 7r/2p) + 0~\ -f-acos [3pt + (f>2. 



Owing to unequal damping, the curves in figs. 19 and 20 

 did not pass through the origin of coordinates as they would 

 have done had the conditions been ideal. 



If, now, the amplitudes are made inversely as the fre- 

 quencies, fig. 19 will become fig. 21, while fig. 20 is replaced 

 by fig. 22. The equations to fig. 21 may be written 



a = a cos [pt + 02 + a/3 cos [3pt + 0] . 



y — a cos [p(t + 7r/2p) + 0)2+ a/3 cos [3p(t + tt/6/>) + </>]. 



This is the well-known caustic by reflexion of parallel rays 

 from the inside of a circle of radins 4a/3. It is the epi- 

 cycloid due to the rolling of a circle of radius a/3 on a circle 

 of twice its radius. 



The equations to fig. 22 are 



x=a cos [pt + 0] +a\Z cos [3p(t -f- ir/6p) + 0] . 



y = a cos [p(t + irj2p) + 0] + a/3 cos [3pt + <£] . 



The curve is a four-cusped hypocycloid, or astroid, traced by 

 a point on the circumference of a circle of radius a/3 rollino- 

 inside a circle whose radius is 4a/3. 



