KECENI PKOGRESS IN HYDE0D"XNAM1CS.- 83 



and the form of tlie wave to the fourth order by 



y + ^ ma^ — §■ m?a* = a cos mx — i ma* (1 + Yk w^*^) cos 2 m:ti 

 + f lii^a^ cos 3 mx — ^ m^ci* cos 4 iJia; .... 

 This embraces so far as the third order the results of the earlier paper. 

 To the third order this agrees with the expansion for a trochoid, and 

 therefore the curve approximates to Gerstuer's and Rankine's form men- 

 tioned below.' In the same paper Stokes also considers the analogous 

 problem for the waves at the common surface between two liquids. When 

 the fluids are confined by horizontal rigid walls there is as before only 

 one form of wave, for given velocity of propagation, and the velocity (c) 

 is given by 2x0- = ^\ (p — p^) \p tanh 2irhj\ + p^ tanh 27r/i,'/A.}"' 

 where h, /t' are the thicknesses of the fluids, and the meanings of the 

 other constants are evident. The case is difiTerent when the upper fluid 

 has a free surface. Here either for given wave-length or for given 

 velocity of propagation there are two possible systems of wave-forms. 

 One of these, when the lower fluid is deep, corresponds to that form, 

 which is propagated on a single surface, and this whatever the depth of 

 the upper fluid. The other form is propagated with velocity given by 

 27rc2 = gX(p- pi) (p tanh 2Trh^ jX + p')"'. 



Only one definite series of waves of permanent typo can be pro- 

 pagated in a fluid in which no vortex motion is present ; but it does not 

 follow that this is the only permanent form which is possible in a perfect 

 fluid. One other at least is known, which was first discovered by 

 Gerstner^ in 1802, and afterwards independently by Rankine'^ in 1862. 

 The latter gives a most elegant geometrical proof that a trochoidal form 

 of wave is a possible one, and that the velocity of propagation is 

 \/((7X/27r). Here the motion of any particle is a uniform one in a circle, 

 the radius of the circle diminishing with the depth. In a later note * he 

 discovers the essential difierence between this mode of wave-motion and 

 that considered by Stokes. It lies in the fact that the exact trochoidal 

 waves possess molecular rotation. Stokes notices this in the reprint of 

 his own papers^ and points out that in order that such waves may be 

 excited in a perfect fluid by operations on the surface alone a preparation 

 must be laid in the shape of a horizontal velocity decreasing from the 

 surface downwards according to the law exp (—4-!rh/\) where k is a 

 function of the depth given by 



depth = 7i; i ^ "~ ^•^-T ( ^ J ( 



The physical interest therefore of these motions is not so great as has 

 been sometimes thought. 



The same objection, amongst others, lies against an attempt by Hagen'' 



' The theory of periodic waves has been treated by Boussinesq in a very long 

 paper in the 3Iem. -par dircrs Samnts, xx. p. 509. He does not seem to have been 

 acquainted with much of the work done outside France. 



^ ' Theorie der Wellen,' Abhand. der Konigl. Bolwiisclwn Gesellscliaft der Wiss. 

 1802. Also printed separately, Prague, 180i. Also in Gilbert's Annalen der Phys-lh, 

 Bd. 32, p. 412. 



^ ' On the exact form of waves near the surface of deep water,' Trans. Roy. Soc, 

 1863. Also Reprint, p. 481. St. Tenant reproduces in the C.R., Ixxi. p. 186, a some- 

 what similar proof by M. Belanger, given by the latter in 1828. 



* Reprint, p. 494. 



* Appendix A to Oscillatory Waves, p. 219. 



" ' Ueber Wellen auf Gewassern von gleichmilssiger Tieie.'—MatJt, Abhand. konig. 

 Ahad. d. Wiss. Berlin (1861), p. 1. This is a long drawn-out paper. 



02 



