304 Prof. G. H. Darwin. On the Correction to the [Apr. 1 , 



This equilibrium law would still hold good when the ocean is 

 interrupted by continents, if water were appropriately supplied to or 

 exhausted from the sea as the earth rotates. 



Since when water is supplied or exhausted the height of water will 

 rise or fall everywhere to the same extent, it follows that the rise and 

 fall of tide, according to the revised equilibrium theory, must be 

 given by — 



iUf^—t («*-»-. (2) 



where a is a constant all over the earth for each position of the moon 

 relatively to the earth, but varies for different positions. 



Let Q be the fraction of the earth's surface which is occupied by 

 sea ; let X be the latitude and I the lougitude of any point ; and let 

 ds stand for cosXdXdl, an element of solid angle. Then we have — 



4>7rQ=^ds 



integrated all over the oceanic area. 



The quantity of water which must be subtracted from the sea, so as to 



depress the sea level everywhere by a«, is 4i7ra s ocQ ; and the quantity 



• • • • SttT/^x^ cos*^^ ^* 



required to raise it by the variable height — — 3 - — _J is the integral 



of this function, taken all over the ocean. But since the volume of 

 water must be constant, continuity demands that — 



Sma 



Ljj(cos 2 s-J)<Zf 



(3) 



integrated all over the ocean. 



On substituting this value of a, in (2) we shall obtain the law of 

 rise and fall. 



Now if X, I be the latitude and W. longitude of the place of 

 observation; h the Greenwich westward hour-angle of the moon at 

 the time and place of observation; and S the moon's declination, it is 

 well known that — 



cos 2 2— ^-=-| cos 2 X cos 2 c cos2(ft— Z)+sin2\sin5 cos 5 cos Qi — l) 



+ Ki-sin 2 £)Q-sin 2 X) (4) 



We have next to introduce (4) under the double integral sign of (3), 

 and integrate over the ocean. 



To express the result conveniently, let — 



