﻿the Duplex Harmonograpli . 505 



the circle of higher frequency having the smaller amplitude. 

 I£ pendulums C and D are set to draw the involute as in 

 fig. 32, while A and B draw a circle in the same direction 

 whose radius is as nearly unaffected by friction as is practi- 

 cable, we may by arranging that the initial radii are as 3:1 

 draw a curve approximating very closely to the limaeon of 

 Pascal. As the ratio of the radii sinks in value owing to the 

 damping of the motions of pendulums C and D, a stage is 

 reached when the shrinking radius is twice that of the circle, 

 and the cardioide is described. Passing through this stage, 

 the decrease in the radius of the slower motion progresses 

 until the radii become equal, which transforms the trace into 

 a trisectrix. The needle was then removed from the plate,, 

 and when all had been reduced to rest a mark was made on 

 the prepared surface by the tracing point. This black speck 

 should be visible in the diagram on the trisectrix at the 

 inner vertex of its loop, but in our picture it is very slightly 

 displaced within the loop. The apses lie on a straight line 

 through the centre. This confirms the view that the motion 

 of the damped pendulums is isochronous through the range 

 covered in the picture. 



The arrangements for fig. 34 are as for fig. 33, except that 

 D is fixed, so that the trace is drawn by a point with a uni- 

 form circular motion of slowly diminishing amplitude to- 

 gether with a simple harmonic motion which is rapidly 

 damped. The frequency of the motion in the circle is twice 

 that of the single vibration. The initial amplitude of the 

 single vibration is three times the radius of the circle. The 

 style was removed when the slower vibration had almost died 

 away, leaving the circular motion alone operative. 



In fig. 35, we have the involute of the circle as in fig. 32 

 combined with a simple harmonic motion due to the vibration 

 of pendulum A, B being clamped. The needle was removed 

 when the damped pendulums and D, drawing the involute, 

 had come to rest. 



Three Frequencies : Two Opposite Pendulums 

 beating. 



The Duplex Harmonograpli lends itself readily to the 

 description of figures in which the combined rectilinear 

 motions are themselves subject to periodic change in 

 amplitude after the manner of beats in acoustics. Of this 

 kind of diagram we shall give a few examples divided into 

 four classes. In the first three of these classes we shall ileal 

 only with three frequencies. In the first, the mean frequency 



