310 DEEP WATEB WAVES, PKOGRESSIVE OK [393 When we retain the third order of small quantities, the equations naturally become more complicated. We now assume that in (3) ft, ft', are small quantities of the second order, and % y', small quantities of the third order. For p, as an extension of (5), we get p = e~y ( - -£ sin SB 4- -£- cos as] + e~zy ( - ~ sin 20 + -j~ cos 2a? ) ^ \ dt dt J \ dt at / cos Su+gy + I- ^ (a2 + a'2) - Ze-w {(a/9 + a'/3') cos a? + (aft' -a ft) sin «} ................... (24) This is to be made to vanish at the surface. Also we find, on reduction, + (1 - 2y) + 2,7/3 sin 2^ - - + 2^' coa 2^ cos 3* - + 4 cos as -j. (aft' + a'ft) + (a2 -f a'2) (a sin ao - d cos x); ......(25) cLt and ab the surface DpfDt = 0 for all values of so. In (25) y is of the form (7), where 5, b', are of the second order, G, c', of. the third order. Considering the coefficients of sin x, cos so, in (25) when reduced to Fourier's form, we see that d2ct/dt2 + get, d*a'/dtz + ga', are both of the third order of small quantities, so that in the first line the factor (1 — y + ^-y2) may be replaced by unity. Again, from the coefficients of sin 2aj, cos 2*, we see that to the third order inclusive and from the coefficients of sin 3a?, cos 3a? that to the third order inclusive And now returning to the coefficients of sin as, cos x, we get ~ + g«- 2a' jt (a? + «>*) + 4 1 (a/3' - a'/3) + a (a4 + ^ = 0, . . .(28) -J? + ^ + 2a ^ (a2 + a'2) - 4 ^ («/?' + a'/S) + a' (a<+ ^) = 0. (29).................. (23)