﻿the Duplex Harmonograpli. 511 



parabola y 2 + ax — 2a 2 — 0. At the beginning of the trace the 

 component rectilinear excursions along the axes diminish T 

 but the motion in the ^-direction dies down more rapidly, 

 and when it is zero its amplitude changes sign so that the 

 subsequently growing parabolas turn their concavities in 

 the opposite direction. The trace continues until the para- 

 bolic motion has dwindled to an oscillation along the axis 

 of x. 



Instead of releasing the pendulums so as to draw parabolas, 

 in fig. 48 they are started so that the successive curves are 

 all examples of Lissajous' figure of eight. The pendulums 

 are released simultaneously, A and B from their outward 

 positions and C and D from their inward positions. As the 

 quivering of the pendulum rods again proved troublesome, 

 the style was placed on the prepared surface when its motion 

 was approximately rectilinear along the axis of y. The 

 equations to the curve are 



,r = acos [2p£ + 7r] +acos {2p + 2h}t, 

 y = a cos [pt + 7r] + a cos {p + h}t, 



the trace lasting from t — Trjh to t — 2irlh. While the motion 

 along the axis of y decreases in amplitude, the motion along 

 the axis of w increases. This motion attains a maximum, 

 and then both begin to die away together and reach zero 

 value simultaneously, when the needle was removed from 

 the plate. By the time that the combined vibration along x 

 had reached its maximum amplitude, frictional damping had 

 notably affected its amount. Otherwise the whole figure 

 would have been enclosed in the space bounded by the four 

 parabolas 



(y±a) 2 ±ax/2-a 2 = 0, 



which also constitute the complete envelope. 



/3. Mean frequencies equal and differences as in a. 



In this case the frequencies will be p + h, p + 2h, p — h> 

 p — 2h. In fig. 49, using these frequencies, with initial 

 amplitudes equal, pendulums A and B are released from their 

 extreme negative position, while and D start with them 

 from the corresponding positive position. The equations to 

 the consequent trace are 



x = acos [{p + h}t + 7r~\ +flcos[|}-//}/, 

 y — a cos \_{p + 2h}l + 7r] +a cos {p — 2Ii}t, 



the record extending from t = 7r/h to t=s2Trjh. The motion 



