Coupled Vibrations elucidated by Simple Experiments. 247 



be visibly coupled, adjusted, and worked, while valuable in 

 themselves as apparatus for dynamical experiments, become 

 doubly so to students striving to grasp the subtler electrical 

 phenomena which make no immediate appeal to the senses of 

 sight and touch but must be imagined at every stage. 



It is in the hope of elucidating the phenomena in question 

 that the present paper is offered. It deals with two kinds of 

 coupled pendulums. These, even when set up very roughly 

 from the materials at hand in any laboratory, serve to yield 

 results suitable for lecture illustration and also afford quanti- 

 tative exercises for the students' practical work. Of course, 

 with more costly mounting results of higher refinement may 

 be obtained. 



As it is difficult to follow the intricacies of the motions 

 while they are being executed, the bobs carry funnels to give 

 sand-traces on a moving board. The paper presents thirty 

 photographic reproductions of these traces, twenty of them 

 showing double traces simultaneously obtained one from 

 each bob of the coupled pendulums whose vibrations are 

 often strikingly different. 



Each type of pendulum shows the change in frequency 

 consequent on gradually varying the closeness of the 

 coupling. They also show the changes in relative amplitudes 

 of the superposed vibrations which follow from the different 

 ways of starting the system. 



II. Equations for Electrical Circuits. 



The electrical case which we wish to represent by a mecha- 

 nical analogy is that of two coupled circuits of negligible 

 resistance each having capacity and inductance. Let us 

 denote the capacities by II and S, the respective inductances 

 by L and N 7 and their mutual induction by M. Further, 

 let the quantities of electricity on the condensers at time t be 

 y and z respectively. Then the simultaneous equations of 

 motion may be written as follows : — 



K £ + s-*S < 2 > 



The coefficient of coupling y is given by 



M 2 



'-IB ^ 



