Laplace's Theory of the Tides. 187 



as Sir William Thomson endeavours to do ; for this equation 

 is completely satisfied by the preceding expression of a with 

 any value of B whatever ; but it must be determined by some 

 condition outside of it ; and this condition is that the volume 

 of the sea must be constant, or, in other words, that the mean 

 depth of the sea must remain the same. This condition is 

 already satisfied for the equilibrium part of this tide repre- 

 sented by H(l — 3fj?) ; and it now only remains to satisfy it 

 with regard to the small part represented by a! . This condi- 

 tion, therefore, will be satisfied with the value of B which 

 satisfies the equation 



\a / da-=0, 



in which cr represents the surface of the globe, and in which 

 the integration must be extended over the whole hemisphere, 

 that is, from fi — to /a = 1. 



Regarding <r as a function of /jl, we have da equal to dp 

 multiplied into a constant ; and hence, instead of the preceding 

 equation, we have 



The integration of the first member of this equation, with 

 the preceding expression of a 1 substituted, gives an expression 

 readily deducible from that of a! by increasing the exponents 

 and subscript members of the constants by unity, and dividing 

 the expressions of the constant by the exponents so increased ; 

 so that it is not necessary here to give the expression. The 

 value of B which makes this expression equal to zero is the 

 value which satisfies the preceding condition, and the value 

 which B must have in the preceding expression of of. 

 With the value of B thus determined, the value of jm in the 

 preceding expression of a! which makes it zero is the cosine of 

 the latitude of the node of the tide, so far as it is represented 

 by d. 



The other of the two arbitrary constants referred to must 



be zero to satisfy the condition that the meridional component 



of motion must vanish at the equator, as is readily seen from 



dd 

 an inspection of equation (2), since by equation (9) -=— 



in equation (2) must vanish at the equator. 



William Ferrel. 



