1915] DEEP WATER WAVES, PEOGRESSIVE OR STATIONARY 307 The velocity-potential <£, being periodic in x, may be expressed by the series <f> = ae-v sin sc — a.'e~y cos x + /30~22/ sin 2cc - fi'e-"y cos 2as + 7 e~*v sin Bos - <y'e~*v cos 3a? + . . . , ... (3) where a, a', & etc., are functions of the time only, and y is measured downwards from mean level. In accordance with (3) the component velocities are given by u = d<f*/dos = e~y (a cos x + of sin as) 4- 2e~22/ (/9 cos 2* -l- /3' sin — y = c£<£/% = e"2/ (a sin a; — a' cos x) + 2e~2^ (/9 sin 2# — $' cos 2#) + .... The density being taken as unity, the pressure equation is p=-d(t>/dt + F + gy~%(u* + vz), .................. (4) in which F is a function of the time. In applying (4) we will regard a, a', as small quantities of the first order, while /3, /?', 7, 7', are small quantities of the second order at most ; and for the present we retain only quantities of the second order. /3, etc., will then not appear in the expression for 11? + v*. In fact and da t . da? d/3 ,, . ,r> = — ^ e~v sin x + -j- e~v cos as — ,- e~w sm r at dt dt + gy~ i-e~2V (a2 + a'2) + F. . . .(») The surface conditions are (i) that p b§ there zero, and (ii) that Dp dp dp dp A fa. i€ = ji + u j + v j - ° ...................... (6) Di dt dx dy The first is already virtually expressed in (5). For the second dp d'*a . • d*-a! ... da J = __6-»am ,+ . rfp da „ cZa' . £ = _. e-y cog x --- a* cw • a^ -r = IT fi^ sin « - ~ e"2' cos « + ... + a + e~^ (a'J + a/a). dy dt dt In forming equation (6) to the second order of small quantities we need to include only the principal term of u, but v must be taken correct to the second order. As the equation of the free surface we assume y = a cos x 4- of sin x + I cos 20 4- 6' sin 2as + c cos 3a; + c' sin 30 + ...... (7) 20—2trata. But when the velocity is high enough, or the viscosity low enough, the motion becomes turbulent, and the flow of heat may be greatly augmented. With the same reasoning and with the same notation as before we have