﻿the Duplex Har monograph. 503 



of a pendulum-bob which contains a gyroscope in rapid 

 rotation *." 



Our apparatus was originally set up in order to illustrate 

 the vibration of a vibrating electron in a magnetic field f. 

 If Ave regard the radiation of: energy from the vibrating 

 -electron as a continuous process, the loss of amplitude is 

 roughly illustrated by the inevitable frictional decrement. 



Fig. 29 results from the suppression of one of the con- 

 stituents of the motion. The equations to the trace are 

 obtained from those to fig. 27 by omitting the second 

 term in the expression for y. We may regard the motion 

 .as being elliptical at any instant, the family of ellipses 

 having the origin as the common centre. The two straight 

 lines y 2 = a 2 , and the two circles (x 2 -\- y 2 ) 2 — 4:d 2 t v 2 constitute 

 the complete envelope of the family. The pen remained 

 down during the same period as in drawing fig. 27. 



Fig. 30 corresponds with fig. 27, but now the component 

 circular motions are in the same direction. In this instance, 

 the pen was placed on the table when the pendulums D and 

 B were released. The simultaneous start of C and A thus 

 occurs when the pen is at ( — 2a, 0). The trace is at first 

 distorted by the vibrations of the pendulum-rods thus set 

 up. The tuning of the pendulums was that employed in 

 fig. 27. The equations to the trace drawn by the four pen- 

 dulums may in this case be written in the more definite form 



x = a cos \_pt + 7r] -\-a cos [\-p-{-li}t-\-7r~], 



y = acos [p{t + 7r/2p) + 7r']+acos [\p+h\(t + ir/2{p + h\) + ir] 9 



where t is counted from the release of C and A, the record 

 lasting slightly longer than ir/h. The innermost loop of the 

 curve is distorted by the backlash due to the play of the 

 jointed parts. The trace is a portion of an epitrochoid traced 

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

 ap/{p + 7i} rolling outside a circle of radius ahj{p-\-7i}. 



Fig. 31 is similar to fig. 27, but now the amplitudes are 

 inversely as the frequencies. The equations to the trace are 



a==a{p + h}/pcos[pt + 0] + acos[.$p + 7i\(t + 7r/2{p + h\) + <l>], 



t/ = a{p + 7i}lpcos[j)(t. + Tr/2p) + 0]+acos[ {p + 7i}t + </>]• 



In this case the duration of drawing is 2tt/7l. The curve 

 is the hypocycloid due to the rolling of a circle of radius a 

 inside a circle of radius {2p-\-h}a/p. The small blank area, 

 at the centre of the figure should have been sensibly circular 

 and of radius 7ia/p. 



* Thomson and Tait, ' Natural Philosophy; Part I. 

 t Wood, ' Physical Optics,' 1011. p. 506. 



