XX. 



LAPLACE'S SOLUTION OF THE TIDAL EQUATIONS. 



Kv 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 : 



y^{l—v^):^^l—v^l^^—{^—2v^—(iv^)u + fiEi'''={) .... (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, i^=sin 3' is the sine of the geo- 

 gi'aphical polar distance 3, Ek^ is the amplitude of the equilibrium tide, 

 and 



I'-l ("' 



n- 1 

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

 ^7 289 



form, in terms of the earth's radius. 

 Putting 



XL=K.y^ ^ K^v^ ^ K^ v^ ff^j/" (3) 



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

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

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



A'2=0. 12^4—12^4=0. 16^6+ /i^=0, etc., 



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

 [Encyclopedia Britannica, 9th ed. art. " Tides," $ 16, vol. xxiil, p. 359. 



ti={K^-E)V^+E,v'+Eeve+..E2iP"-' . . . (34.) 



3L) 



