﻿the Duplex Harmonograph. tjOl 



being 2a cos lit cos [pt + it] , the instantaneous value of the 

 amplitude may be regarded as 2a cos ht, whilst the oscilla- 

 tions are always in quadrature with the motion along y. 

 The trace is the projection of the motion of a point in a 

 circle of radius 2a with uniform angular velocity p, the 

 circle rotating about a diameter with uniform angular 

 velocity Ji. Similar methods of representation are applicable 

 in other cases. Frictional decrement can likewise be repre- 

 sented on this view. 



Fig. 37 was produced under identical conditions with the 

 exception that the motion of B was suppressed. It may be 

 regarded as the projection of fig. 36 on a plane at 60° to it 

 through the axis of x. 



Fig. 38 is produced by altering the conditions of release 

 from those of fig. 36. The bobs of all the pendulums start 

 from their outward positions simultaneously. The equations 

 to the trace are 



,^ = acos \\p + h}t + 7r] -f a cos [{p — h}t-\-Tr], 

 y = a cos \_pt -f 7r] -fa cos \_pt -f ir\ , 



so that the displacements along x and y pass through their 

 zero values simultaneously. The trace lasts from t = ir\h to 

 t = 27r/Ii. In this time the amplitude of the approximately 

 rectilinear simple harmonic motion changes from its maximum 

 value 2 \/2a, passes through its minimum 2a, and again 

 attains its maximum. During this interval the direction of 

 the resultant motion changes through a right angle, and 

 when t = 2irlli the motion is again at 45° to the axes. The 

 amplitude of the motion along the axis of x now begins to 

 decrease, and the direction of the resultant rectilinear vibra- 

 tion rotates in the opposite direction. This is illustrated in 

 fig. 39, in which the oscillation of the direction of the re- 

 sultant vibration can be readily traced owing to the loss of 

 amplitude by friction as the oscillation takes place. The 

 equations to the trace being as before, the record in tig. 39 

 lasts for the time 2irlh from t—ir\2h. If friction had been 

 inoperative both diagrams would have been bounded by 

 y=±2a. 



The other Frequency half their Mean. 



The pendulums are now tuned so that the third frequency 

 is half that of the mean frequency of the pair of oppositely- 

 placed pendulums which produce beats together. 



In fig. 40 the frequencies are in the ratios 2p-ir7t, p-j-h, 

 2p + 3A, p + h. All the amplitudes are equal, and the pen- 

 dulums are released tooether from their extreme outer 



