Theory of the Tides 273 



a level surface under the combined action of gravity, and of the disturbing 

 force. The sum of the potentials of these forces must be constant at this 

 surface. Then no work will be performed when a particle of water is moved 

 along the surface. This can also be expressed by saying that the pressure force 

 and the tide generating force must be in equilibrium at any point. If r\ denotes 

 the elevation of the water above the undisturbed level, then gravity potential 

 is grj, and the above condition requires that grj+Q = constant = C, or 



tj = -- + C 



g 



when Q denotes the disturbing potential. 



If we put mR/g = H, we obtain with (VIII. 4) 



rj = #(cos 2 #-|) + C, (IX. la) 



in which M = mass of the moon; a = distance moon-centre earth; E = mass 

 of the earth; R = earth's radius, & = moon's zenith distance. 



The equilibrium form of the free ocean surface is a harmonic spheroid 

 of the second order, of the zonal type, whose axis passes through the dis- 

 turbing body. If we consider first a disturbing body alone, its equilibrium 

 flood yj attains a maximum for & = 0° and 180°, i.e. when the disturbing 

 body is in the zenith or the nadir and V] max = \H and a minimum for 9 = 90°, 

 which is when the disturbing body is on the horizon and ?? min = —\H\ the 

 amplitude of the oscillation is, therefore H. 



If we put R = 6370 km, we have 



lunar tides solar tides combined effect of both 

 0-55 nT 0-24 m~ _ 0-79 m 



Owing to the diurnal rotation of the earth and to the orbital motion of 

 the disturbing body, there are periodic variations in rj for a given point of 

 the earth's surface; their periods are exactly the same as those* of the tide 

 potential discussed on page 263. We have for rj the same developments as 

 in (VIII. 13); only we now have to substitute \H for A. Consequently, there 

 are three kinds of partial tides in the equilibrium theory: long period, diurnal 

 and semi-diurnal tides, and all that was discussed on the subject that the 

 harmonic analysis of the tide potential can be applied without restriction 

 to the partial tides of the equilibrium theory. The different terms constituting 

 the tide-potential Q can be expressed in the quantity ij, and this is the usual 

 form in which it is presented. 



We will discuss the combined action of the moon and the sun in connec- 

 tion with the semi-diurnal tides. If we introduce in the equation (VIII. 13) 



18 



