﻿the Duplex Harmonograpli. 513 



In fig. 50 the combined oscillations o£ the oppositely- 

 placed pendulums are always in the same phase, while in 

 tig. 51 they are always in quadrature. The bob of pendulum 

 C was started from its inward position when the others were 

 released from their outward positions. A and C are thus 

 in opposition, while B and D conspire. The equations to 

 the trace are 



ar=acos [{2p + h}t + tt]+ a cos {2p + 3h}t, 



y = a cos [{t> + 2/i}£ + 7r] -\-a cos [p£ + 7r], 



from t = 7r/'2Ji to t = 7r/h. In this case the successive curves- 

 are still Lissajous' figures of eight. The process of passing 

 from a straight line to another at right angles by transfor- 

 mation of the shape of the curve occupies the time tt/2/a- 

 All the curves lie within the area enclosed by the parabolas 



y 2 ±2ax-±a? = 



which constitute the envelope. 



Returning again to the condition that the resultant vibra- 

 tions along the two axes shall be in the same phase, in 

 fig. 52 we set these resultant, motions to draw a parabola 

 instead of a Lissajous' figure of eight, ys in fig. 50. To do 

 this the initial amplitudes of the two combined motions are 

 maxima, the equations to the trace being 



A'--acos [{2p + h}t + ir] +a cos [{2p + 3/i}£ + 7r] t 



y = acos \_{p-\-21i}t-\-ir\ +acos \jpt-\-w], 



from t = 7r/2Ji to t = 7r/h. The style was removed when the 

 maximum parabola was being drawn. If the needle had 

 been allowed to continue tracing, its parabolic path would, 

 have gradually decreased in size until the point had readied 

 the origin. The subsequent trace would have consisted of a 

 series of growing parabolas drawn with the vertex in the 

 opposite direction, the whole cycle of operations involving 

 the drawing of the two families of oppositely-placed para- 

 bolas, each consisting of an increasing and a decreasing 

 series. 



This is illustrated in fig. 53, in which the conditions arc 

 the same except that the tracing commences with the release 

 of the pendulums. The path of the pen is at first distorted 

 by the elastic vibrations of the pendulum rods. The de- 

 creasing half of the first family of parabolas and the in- 

 creasing half of the second family arc shown on the trace, 

 the time occupied being ir/h, half the period of the complete 



Phil. Mag. 8. 6. Vol. 2D. No. 172. April 1015. 2 L 



