1886.] Dynamical Theory of the Tides of Long Period. 339 



is such that the modulus of decay is short compared with the semi- 

 period of oscillation. 



The quickest of the tides of long period is the fortnightly tide, hence 

 for the applicability of Laplace's conclusion, the modulus of decay 

 must be short compared with a week. Now it seems practically 

 certain that the friction of the ocean bed would not much affect the 

 velocity of a slow ocean current in a day or two. Hence we cannot 

 accept Laplace's hypothesis as to the effect of friction. 



We now, therefore, proceed to the solution of the equation of 

 motion when friction is entirely neglected. 



The solution here offered is indicated in a footnote to a paper by 

 Sir William Thomson (' Phil. Mag.,' vol. 50, 1875, p. 280), but has 

 never been worked out before. 



The symmetry of the motion demands that u, when expanded in 

 a series of powers of ja, shall only contain even powers of fi. 



Let us assume then 



Then 



dfjL 



[^ 2 J]=£i+3(-B 3 -W+ ••• +(K+1)(1W-W+ 



.... (5) 



Again 



J = -pB^ + (B l -fB 3 )^+ . . . +(I?«- I -/3IW^ m + • • • 



« = i)-iftB 1 '+i(B l -f*Bjt4 . . . +i(B 2i _ s -r~B,j_W+ . . , 



... (6) 



where C is a constant. 



Then substituting from (5) and (6) in (4), and equating to zero 

 the successive coefficients of the powers of /», we find, 



C=-lE+B 1 lp 



B s -B 1 (l-lfl3) + ^E = I 



