Vibrations elucidated by Simple Experiments. 257 



Putting these in (36)-(39) we obtain equations which are 

 satisfied by 



Hence the corresponding special solution may be written 

 b ± , 1 + /3, mt /k «n 



^=-JT^ cosmt+ 2T/3 bcos Va^y •/ (o2) 



b 1 + /3, mt /Ra . 



t cos mt+- ^ocos /n ^-. . . (Do) 



2 + /3^" 2+f3 v ™ \/(l + l3y 



Accordingly the ratio of amplitudes of quick and slow 

 vibrations in the y and z traces are respectively 



+ l:(l + /3) (54) 



(iv.) Cases of Single Frequency. — It may be seen imme- 

 diately that if the phases are opposite and amplitudes equal 

 the strut CC and the bridles ACA and A'C'A' will all 

 remain at rest. We should accordingly have the quick 

 vibrations alone, which are proportional to cos mt. Again, 

 if the phases are alike and the amplitudes equal we should 

 have both pendulums swinging in unison, each bridle and 

 its lower thread forming one plane. The vibrations are 

 accordingly of the slower type of period ^/(l + /3)-times the 

 other just noticed. 



Y. Theory of Cord and Lath Pendulums. 



Description. — The model here called the cord and lath 

 pendulum consists essentially of two pendulums PR and QS, 

 one suspended by cords from a movable point R on the lath 

 of the other as shown in fig. 3. The bobs are shown by P 

 and Q, the movable and fixed points of suspension by R and S. 

 The cord pendulum is shown by PR. When the apparatus 

 is used for giving double sand-traces a system of four cords 

 and three stretchers is used so as to clear the upper board 

 which takes the trace from the bob Q. Each cord is pro- 

 vided with a tightener for adjusting its length. Of the two 

 upper stretchers the back one is shown in the figure a little 

 shorter and lower so as to make it and its cords visible. In 

 the actual model the cords and stretchers are all symmetrical. 

 The point of suspension R is adjustable on the lath of the 

 pendulum QS by a small sliding metal sleeve with studs to 

 support the cords. This sleeve is set where desired and then 

 fixed in position by a screw-clamp. The suspension S of the 



