Deep-sea Tides, and the Effect of Tidal Friction. 183 



But the equations reducing themselves in this case to 



2(B— gc) cos 20==2n sin 6 cos 6v l — mu li 

 — 2(B — gc) cos = m sin 0v x — 2n cos 0u l} ] 

 . oa dfc , Cdu. ' cosO ~] 



the equations for determining w x and v 2 become 



2(B —^rc) {w cos 20 + 2w cos 2 0} = {4in* cos* 0—m*\u lf 

 2(B— ^c) {2rc cos 20 + m] cos = {4n 2 cos 2 0— w 2 } sin 0v y \ 



m 



and the right-hand coefficients vanish when cos0= + — ; and 



our approximations fail whenever m < 2n, unless the left-hand 



coefficients vanish at the same time. 



in 

 Now, giving cos 2 the value — -gj these coefficients will be found 



to be 



vn 1 



K—^\rn^-\-mn^2n l \ i and -{m 2 + m?i— 2n 2 } , 



both of which vanish when m=n. This case, therefore, of which 

 alone Laplace has treated, is no longer, as before, at the boun- 

 dary between possible and impossible conditions, but a singular 

 possible case in the midst of impossible ones. 



And it is singular in its algebraical no less than in its physi- 

 cal character; for the differential equation determining k loses 

 its general form, and consequently the position of the wave no 

 longer depends on the equatorial depth. 



For making m=n, we find 4 cos 2 — 1 is a common factor of 

 each of the equations for u x and v v and they become 



2(B— gc)=nu l (whence -^=0], 



and 



2(B— gc) cos 0=nv x sin ; 



whence the coefficient of tc in the equation of continuity vanishes, 

 and it becomes 



die n*c . Q 



the integral of which may be written 



n 2 ccos 2 7i . ' , 



.* aa/+ 2(B=^ a=/ 1 1 -fi rcos $ h 



