ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 173 



It follows that if the disturbing potential be of order s + 1 and rank s, and the 

 period be ^ (* + 1) days, the tide will involve no rise and fall at the free surface, but 

 will consist merely of horizontal currents. If we put s = I the requisite period will 

 be rigorously equal to a sidereal day, and the circumstances will correspond with those 

 we have assumed to characterize the solar diurnal tides. We therefore conclude 

 that in an ocean of uniform depth the solar diurnal tides will involve no rise and fall. 

 We shall however see hereafter that for certain of the lunar diurnal tides the 

 difference between the period and a sidereal day may be sufficiently great to render 

 the rise and fall of considerable importance, unless the depth is very small. 



We have seen in 4 that the formulae applicable to the case of variable depth may 



be deduced from those applicable to the case of uniform depth by replacing h by K 



i P > -~i 



and C; by Q, + -^ \n (n + 1) + - r*,. 



-id) "fi" I A, 



L. 1 



But if y* = 0, r", = g,,C*,, and therefore when X 2w/(.s -f- 1) and all the y'.s 

 are zero except yl+i, Cj +3 , Cj +6 , &c., will all be zero, while 



whence we obtain 



c: +I =- . 2 ^.---^-.. 



1 -2(s + 1)- %\, 

 Thus the forced tide will be similar in type to the disturbing potential which 



produces it, though it will be inverted unless I < or > .- 



(s + l)-/, +1 



This theorem will admit of application to the solar diurnal tides on putting s = 1 , 

 in which case it reduces to a theorem given by LAPLACE.^ The critical value of /, 

 for a system comparable with the earth, is considerably greater than such depths as 

 occur on the earth ; hence, for depths comparable with that of the ocean, the diurnal 

 tides will be inverted when the ocean is deeper at the equator than at the poles : 

 they will however be direct when I is negative, so that the ocean is deeper at the 

 poles than at the equator. 



It is evident that the equations typified by (31) will all be satisfied with the Cs 

 all zero if h = 0, since in this case the right-hand members will reduce to zero. In 

 like manner the corresponding equations which apply to an ocean of variable depth 

 will all be satisfied when K = 0, if for all values of n 



* LAIIB, 'Hydrodynamics,' 212. 



