Lord Rayleigh on Waves. 279 



the summation being extended to all the admissible values of 

 k. The value of n 2 is given by 



n =9 K ^T— 



e M + e~ Kt 



If the system be at rest at£ = 0, and displaced according to the 

 law h = r cos (that is, with an inclined plane surface), the sub- 

 sequent motion is given in rapidly converging series by 



, 2J 1 (* 1 ?') cos 9 2J 1 (/e 2 r) cos (9 /VN 



P.S. — Some recent observations on the periods of the oscil- 

 lations of water in a large circular tank may be worth record- 

 ing. The radius of the tank is 60 '3 inches, and the depth 

 about 43 inches. The oscillations were excited by dipping 

 one or more buckets synchronously with the beats of a metro- 

 nome set approximately beforehand. Soon after the with- 

 drawal of the buckets the vibrations were counted (in most 

 cases for five minutes), and the results reduced for a space of 

 one minute. 



Gravest symmetrical mode. — -Frequency by observation 47*3. 

 The theoretical result for an infinite depth is 47*32, and for 

 actual depth 47*13. 



Next highest symmetrical mode. — By observation, frequency 

 = 64*1, by theory 64*02. In this case the correction for finite 

 depth is insensible, and the length of the equivalent pendulum 

 = R-h7*015. 



Gravest mode with one nodal diameter. — By observation, 

 frequency = 30*0. Bv theory, for infinite depth 32*81, for actual 

 depth 30*48. 



One nodal diameter and one nodal circle. — By observation, 

 frequency =56*0, by theory 55*8. The length of equivalent 

 pendulum = 11-*- 5*332. 



Two nodal diameters. — By observation, frequency =41*5. 

 By theory, for infinite depth 42*09, for actual depth 41*59. 



The agreement between theory and observation is as close 

 as could be expected. 



I have lately seen a memoir by M. Boussinesq (1871, 

 Comptes JRendus, vol. Ixxii.), in which is contained a theory of 

 the solitary wave very similar to that of this paper. So far as 

 our results are common, the credit of priority belongs of course 

 to M. Boussinesq. 



