Lord Rayleigh on Iso-periodle Systems. 567 



the potential of the machine and not require a spark-gap; it 

 should be roughly suitable for a coil-spark of 6 in. to 14 in., 

 according to the size of machine with which it is excited. 



The illumination produced will then be good and steady, 

 and the tube may be run for an almost indefinite period 

 without running the slightest danger of over-heating its 

 terminals or of being troubled by its resistance changing. 



A pair of 19-in. disks is adequate to show brightly the 

 bones in the hand and arm, and, with some people, to faintly 

 indicate the ribs on a screen ; while a pair of 12-in. disks 

 exciting a suitable tube is sufficient to show the hand- and 

 wrist-bones clearly. 



LXII. On Iso-periodic Systems. By Lord Kayleigh, F.R.S.* 



IN general a system with m degrees of freedom vibrating 

 about a configuration of equilibrium has m distinct 

 periods, or frequencies, of vibration, but in particular cases 

 two or more of these frequencies may be equal. The simple 

 spherical pendulum is an obvious example of two degrees of 

 freedom whose frequencies are equal. It is proposed to point 

 out the properties of vibrating systems of such a character 

 that all the frequencies are equal. 



In the general case when a system is referred to its normal 

 coordinates fa, fa, . . . we have for the kinetic and potential 

 energies f , 



V = ic 1 1 2 + ic 2 2 2 + .../' • ■ ■ ■ W 

 and for the vibrations 



fa = A cos {n x t — a), 2 = Bcos {n 2 t— /3), • • (2) 

 where A, B, . . . a,, (3 . . . are arbitrary constants and 



n 1 2 =c 1 /a 1 , n 2 2 = c 2 /a 2 , (3) 



If n l9 n 2 , &c, are all equal, T and Y are of the same form 

 except as to a constant multiplier. By supposing a, /3 . 

 equal, we see that any prescribed ratios may be assigned to 

 fa, (/> 2 . . ., so that vibrations of arbitrary type are normal 

 and can be executed without constraint. In particular any 

 parts of the system may remain at rest. 



If x, y, z be the space coordinates (measured from the 

 equilibrium position) of any point of the system, the most 

 general values are given by 



x = Xi cos nt + X 2 sin nt ~) 



y = Y 1 cos nt -f Y 2 sin nt > , . . . . (4) 



z= Zx cos nt 4- Z 2 sin nt J 



* Cominunicated by the Author. 



t See, for example, ' Theory of Sound,' § 87. 



