XX. 



LAPLACE'S SOLUTION OF THE TIDAL EQUATIONS. 



I>v William Ferrel. 



In this paper (supplementary to that under the same heading in 

 vol. ix, No. 6, of the Astronomical Journal), it is proposed to explain 

 more fully a certain point in the latter (which did not appear clear to a 

 correspondent some time since), by presenting the matter more in detail, 

 and also to clear up some doubts held by some with regard to theconver- 

 gency of the series in the tidal expression. 



In Darwin's Equation No. (34), t we have the following differential 

 equation to be satisfied, which is equivalent to that of Laplace : 



v i (l_ v 2)_^J __ v ^—(8— 2k 2 — fiv*) u+ft Ev 6 =0 .... (1) 



dV dV [Darwin's Eq. (33).] 



in which u is the difference between the real amplitude of the tide and 

 that given by the equilibrium theory, r==sin 3" is the sine of the geo- 

 graphical polar distance 3, Ev z is the amplitude of the equilibrium tide, 

 and 



gl 



n 2 1 

 in which '- = -—— and I is the depth of the ocean, supposed to be uni- 

 g 289 



form, in terms of the earth's radius. 

 Putting 



u=K 2 v 2 + K^ + K 6 v 6 K u v» (3) 



in which n is any even number, corresponding with the exponent, and 

 substituting this value of u and its derivatives in (1) above, we get, 

 by equating the coefficients of like powers of v to 0, 



K 2 =0. 12iT 4 — 12^4=0. 16K 6 +fiE=0, etc., 



* From Gould's Astronomical Journal, 1890, vol. x, pp. 121-125. 

 t Encyclopedia Britannica, 9th ed. art. " Tides," § 16, vol. xxm, p. 359. 



u=(K--E) v' 2 +K 4 vt+Ke, v e + . . K 2i v* . . . (34.) 



31 J 



