338 



Prof. G. H. Darwin. On the 



[Nov. 25, 



where m— — • 



9 



Whence x{ f^-co^e) = ± % 



KJ J 4ra de 



• a /M m\ 1 cos 6du 



Then substituting for % and y in (2), which, when 7 is constant 

 and 7] is not a function of 0, becomes 



fta + -r^- -^-(£sin 0)=0, 



This is Laplace's equation for tidal oscillations of the first kind.* In 

 these tides / is a small fraction, being about 2V in the case of the 

 fortnightly tide, and e the coefficient in the equilibrium tide is equal 

 to cos 2 0), where E is a known function of the elements of the 



orbit of the tide-generating body, and of the obliquity of the ecliptic. 



If now we write y3=4ma/7, and /t=cos 0, our equation becomes 



iMih^^-^ ■ ■ ■ w 



In treating these oscillations Laplace does not use this equation, 

 but seeks to show that friction suffices to make the ocean assume at 

 each instant its form of equilibrium. His conclusion is no doubt 

 true, but the question remains as to what amount of friction is to be 

 regarded as sufficing to produce the result, and whether oceanic tidal 

 friction can be great enough to have the effect which he supposes it 

 to have. 



The friction here contemplated is such that the integral effect is 

 represented by a retarding force proportional to the velocity of the 

 fluid relatively to the bottom. Although proportionality to the square 

 of the velocity would probably be nearer to the truth, yet Laplace's 

 hypothesis suffices for the present discussion. In oscillations of the 

 class under consideration, the water moves for half a period north, and 

 then for half a period south. 



Now in systems where the resistances are proportional to velocity, 

 it is usual to specify the resistance by a modulus of decay, namely, 

 that period in which a velocity is reduced by friction to e -1 or 

 l-i-2' 783 of its initial value ; and the friction contemplated by Laplace 



* ' Mecanique Celeste.' 



