﻿508 Dr. J. H. Vincent and Mr. G. W. Jude on 



positions. The needle is allowed to trace from the com- 

 mencement of motion during half the cycle of operations, so 

 that the equations to the trace are 



x = acos [{2p + h}t + ir\ + a cos [{2p + M}t + ir] 3 

 y = acos [{p + h}t + 7r'] +a cos \_{p + h}t + 7r~\, 



from t—0 to t—ir/h. The trace commences with the 

 parabola y 2 -f ax— 2a 2 = 0, and if we neglect the effects of 

 friction all the succeeding parabolas may be derived from 

 this by altering the scale of drawing and the sign of the 

 ^'-coordinate. The points where all the parabolas cross the 

 axis of y are (0, ±a\/2), coinciding with the cusps of the 

 envelope of fig. 54. Since the vibration along the axis of y 

 has a sensibly constant amplitude, the figures will be bounded 

 by the lines y — ±2a and the parabolas y 2 ±ax — 2a 2 = 0. 



In the introduction we state that the figures may be re- 

 produced with fidelity. This extends even to cases in which 

 the length of record varies, as the frictional loss is almost 

 the same whether the needle is in contact with the smoked 

 glass or not. This is illustrated in fig. 41, in which the 

 second quarter of the complete cycle of operations is 

 shown. 



In fig. 42 we are again only concerned with three different 

 frequencies, these being identical with those -of figs. 40 

 and 41. Pendulum C was in this case released with its bob at 

 the extreme inside position, all the other initial displacements 

 being negative. On release the motion consists of a sensibly 

 rectilinear simple harmonic motion along the axis of y. This 

 changes into motion in Lissajous' figure of eight. The style 

 was placed down when the amplitude along the axis of x had 

 first attained its maximum. The equations to the trace are 



.i' = acos [{2p + h}t + 7r~\ +acos {2j) + 3h}t, 



y = acoa [{p + ]i}t + Tr] +acos [{p + h}t + 7r], 



from t = ir\2k to t = 7r/h. 



The other Frequency twice their Mean. 



The remaining examples we shall give involving three 

 frequencies have the third frequency twice the mean of those 

 of the oppositely-placed beating pendulums. 



In fig. 43 the frequencies are in the ratios 2p + h, p + 1i, 

 2p + h, p. The amplitudes are equal, and the pendulums 

 start together with their bobs displaced outwards. The pen 



