Theory of the Titles 283 



assumption, the equations have a solution and we can calculate the forced 

 oscillations, if the depth of the ocean follows the law h — /j (1 — ccos 2 #), 

 where c is any given constant. From this follows that, if the water depth is /?„ 

 at the equator, it will be /7 n (l — c) at the pole. We obtain by simple compu- 

 tation 



IchlmR 8c , _ 4/7? R 



n = -\ J , , n V = — t, — k~ V when 6 = . 

 ' \ — IchlmR ' 8-$c ' ' h 



This relation shows first of all that the ratio of the tidal ranges is everywhere 

 constant in relation to those given by the equilibrium tide; further, that 

 ry = 0, when c = 0, i.e. in the case of uniform water depth, there are no diurnal 

 tides in the ocean so far as the rise and fall of the surface is concerned. How- 

 ever, although there is no rise and fall of the surface, there are, nevertheless, 

 diurnal tidal currents. Laplace derived this remarkable result from his theory, 

 and considered it very important, in view of the fact that the diurnal partial 

 tides are almost completely absent in the European waters; in fact, he wanted 

 to show by this that the dynamic theory is able to do away with the discrepancy 

 between the observations and the results of the equilibrium theory. However, 

 the diurnal tides having proved to be large at many points of the ocean, this 

 conclusion had to be discarded. 



With the oscillations of the third species (s = 2), which include^ the lunar 

 and solar semidiurnal tides, the disturbing potential of the tidal force is 

 a sectorial harmonic of the second order, hence 



r\ = A 3 sm 2 dcos{ot J r 2?. + e) 



in which a is nearly equal to 2co. The computation can be considerably sim- 

 plified, assuming that the orbital motion of the disturbing body is very slow, 

 so that a = 2(» and therefore /= 1. This approximation is rough for M 2 , 

 but there is a "luni-solar" semi-diurnal tide whose speed is exactly 2co if we 

 neglect the changes in the planes of the orbits. If here the depth is assumed to 

 change according to h = h s'm 2 &, (at the equator the water depth // , at the 

 pole zero) there is a solution 



8 _ 2h/mR - 



n = /3-8 l/= l-ilh/mR) 11 ' 



In this case too, the ratio of the tidal range to that of the equilibrium tide 

 is constant, but as /i > 10 for depths existing actually in the ocean, the 

 tide is everywhere inverted, i.e. there where the equilibrium tide is high, low 

 water occurs, and vice versa. It should be noted that at the pole the depth 

 becomes zero (& = 0) and consequently the velocity at the pole becomes 

 infinite. 



Laplace has also derived the semi-diurnal tide (s = 2,f= 1) for a sea of 

 uniform depth. In this case, there are again continued fractions for the de- 



