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



at any instant may be regarded as approximately a simple 

 harmonic motion in a straight line. The direction of vibra- 

 tion in this line oscillates through an angle 2 tan -1 2 ; when 

 t = 7r/h the resultant amplitude is zero and its direction makes 

 an angle tan -1 ( — 2) with the positive direction of the axis 

 of x. This angle increases until, when the motion along y 

 is zero (i. e. when t = 37r/2h) the maximum amplitude in the 

 direction of x equals 2a. In its growth from zero the 

 amplitude passes through a maximum value of oa/'2 when its 

 direction of motion makes an angle tan -1 (— n/3/2) with the 

 -axis of x. If we neglect the decrement in the oscillations 

 due to friction the excursions of the tracing-point are limited 

 by the Lissajous' figure of eight 



{ 



x = 2a sin ht, 



.f/ = 2a sin 2ht, 



or 



x 2 -4:aV-4a 2 y 2 =0. 



<y. Mean frequencies as two to one and differences equal. 



The remaining examples of figures drawn by four pen- 

 dulums in which the opposite pendulums are beating, all 

 obey the conditions that the mean frequencies of the opposite 

 pairs are as two to one and that the differences are equal, so 

 that the frequencies may be taken proportional to 2p + 7i, 

 p + 2h, 2p + 3h,p. Using three frequencies only, we have 

 shown how families of Lissajous' figures of eight may be 

 drawn in which succeeding members increase or decrease 

 in one dimension, the curves in figs 44 and 42 degenerating 

 into two straight lines respectively at right angles. In fig. 50 

 both these effects go on simultaneously and at the same rate. 

 To draw the figure the pendulums were started together 

 from positions of equal displacement, A and B outward and 

 and D inward. The needle was put down when the com- 

 bined amplitudes were at a maximum, the record extending 

 from t = 7r/2h to t = 7r/h, the equations to the trace being 



t y = acos [{2p + h}t + Tr'} +acos {2p + 3Ji}t, 



y=-a cos \_{p + 2h}t + 7r~\ + a cos j)t. 



The complete trace would contain two families of similar 

 and similarly-placed curves, each family consisting of two 

 series in which the curves are increasing and decreasing 

 respectively in amplitude, the curves in each family being- 

 described in contrary directions. 



