﻿the Duplex Harmonoyraph. 501) 



traces during a quarter of a complete cycle. The equations 

 to the trace are 



x=acos [{2p + h}t + 7r] +a cos [{2p + h}t + 7r], 



y = a cos \_{p + h}t + t n-~\ + a cos [ptf + 7r] , 



from t = 27r/h to t = 37r/h. On release, the motion is along 

 the parabola with the maximum parameter. The component 

 of this motion along the axis of y dies down to zero and then 

 increases to a maximum, when the original parabola is again 

 traced. At this stage, the pen was permitted to record and 

 the trace was stopped when the motion had again become 

 sensibly rectilinear. 



We have had an example in fig. 42 in which motion in a 

 Lissajous' figure of eight degenerates into rectilinear motion 

 along one of its axes of symmetry. In fig. 44, the motion 

 becomes rectilinear along the other axis of symmetry. The 

 initial amplitudes of the four pendulums are equal while the 

 frequencies again fulfil the conditions of fig. 43, but h is 

 much smaller compared with p, so that the effect of the 

 frictional decrement in the amplitude of the motion along x 

 is very marked. A and C release B and D in mid-swing, 

 the bobs all starting from their outer positions. Counting- 

 time from the instant when B and D commence to swing, the 

 equations to the trace are 



x=acos [{2p + h}t+3ir/2] +acos [{2p + A}* + 3w/2], 



y = a cos \_{p + li}t-\-ir'] +acos [p£ + 7r], 



in which the frictional decrement is neglected. The record 

 lasts until t — ir\h. Instead of the figure being bounded by 

 the two straight lines x= + 2a its central portions are sensibly 

 contracted, so that when the motion is reduced to oscillation 

 along x its amplitude is no longer 2a. 



It the trace had been continued, this straight line would 

 have opened out again into the figure of eight. This is 

 shown in fig. 45, in which the trace commences when t = 7r/h. 

 A corresponding frictional effect is again evident. The 

 record again lasts for the time nr/h. In fig. 4G a similar 

 cycle of events is recorded, the pendulums ha vino- been 

 slightly readjusted. In this case both the waning and the 

 waxing in the ^-direction are recorded, the time occupied 

 being 2irjlu 



