﻿tlie Duplex Harmonograph. 497 



Frequencies as 1, 2, 3, 4. 



Initial displacements all negative. 



Figs. 6 to 9 form a similar group of diagrams to that 

 discussed in the previous paragraph, the frequencies being 

 now proportional to the numbers 1, 2, 3, and 4, while pen- 

 dulums B, C, and D are fixed in turn in each of the figures 

 7 to 9. The curves in each case represent one half the time 

 required to complete a cycle of operations, the appropriate 

 equations being derived from those of the previous paragraph 

 by substituting 1, 2, 3, and 4 for 2, 3, 4, and 5. If pen- 

 dulum A is clamped the other three draw a diagram identical 

 with fig. 5 turned anti-clockwise through a right angle. 



Various initial conditions. 



The conditions are unaltered except that A is started from 

 its extreme position of positive displacement. 



The equations to the trace in fig. 10 are therefore 



x = a cos pt -fa cos \?>pt + 7r] , 



y = a cos [2pt + 7r] + a cos [4p£ + w] . 



Figs. 11 and 12 are drawn with pendulums B and D fixed 

 in turn. The equations corresponding to these cases are 

 found by deleting the first and second terms respectively in 

 the expression for y. 



In figs. 13, 14, and 15 the initial displacement of B is 

 positive, while the other pendulums start with their bobs 

 in the outer position . The amplitudes are equal and the 

 pendulums are released simultaneously. The equations to 

 the curve shown in fig. 13 are 



x = a cos \_pt + 7r] + a cos [3pt -f 7r] , 

 y = a cos 2pt -\- a cos [4p£ -t- 7r] . 



The equations to the curves 14 and 15 are found by 

 omitting in turn the first and second terms in the expression 

 for x. 



In fig. 16 pendulums A and B are started from the in>ide 

 position, while C and D commence from the outside. The 

 equations to the trace are thus 



x = a cos pt + a cos \_3pt -f ?r] , 

 y = a cos 2pt -f a cos [4^)£ + tt] . 



To draw fig. 17 we employ the methods described in the 

 paragraph headed " Initial Amplitude and Phase" to control 

 the circumstances of release. The amplitudes and periods 



Phil. Mag. S. G. Vol. 29. No. 172. April 1915. 2 K 



