Mr. stokes, on THE THEORY OF OSCILLATORY WAVES. 455 



AVhen the depth of the upper fluid is very small compared with the length of a wave, one 

 of the roots of (t6) will be very small ; and if we neglect square and products of mh and Yi the 

 equation becomes 2 pD'(^- 2 (p - p) mli^D = 0, whence 



^ . P^P'^nh,, c^=P~P' gh, (47). 



P P 



These formula; will not hold good if mh be very small as well as m/i^, and comparable with it, 

 since in that case all the terms of (46) will be small quantities of the second order, mh^ being 

 regarded as a small quantity of the first order. In this case, if we neglect small quantities of the 

 third order in (l(j), it becomes 



4(0^- - -iw/p (A + h) i[+ -iip - p) ni'hh^ = 0, 



whence c- = ^ |/, + /(^ ± /sj (]i - hy- + tPj.hh\ (48). 



Of these values of c"-, tliat in which the radical has the negative sign belongs to that system of 

 waves to which the formula; (47) apply when h^ is very small compared with h. 



If the two fluids are water and mercury, — is equal to about 13.57. If the depth of the 



P. 

 water be very small compared both with the length of the waves and witli the depth of the 



mercury, it appears from (47) that the velocity of propagation will be less than it would have 

 been, if the water had rested on a rigid plane, in the ratio of .9624 to 1, or 26 to 27 nearly. 



G. G. STOKES. 



Vol.. VIII. 1'akt IV. 3N 



