EQUILIBRIUM THEORY. 365 



Expanding in powers of r/D t which is in our case a small quantity, and 

 retaining only the most important term, we find 



Considered as a function of the position of P, this is a zonal harmonic of the 

 second degree, with OC as axis. 



The reader will easily verify that, to the order of approximation adopted, 

 G is equal to the joint potential of two masses, each equal to \M^ placed, one 

 at C, and the other at a point C' in CO produced such that OC' = OC*. 



b. In the ' equilibrium-theory ' of the tides it is assumed that the free surface 

 takes at each instant the equilibrium-form which might be maintained if the 

 disturbing body were to retain unchanged its actual position relative to the 

 rotating earth. In other words, the free surface is assumed to be a level- 

 surface under the combined action of gravity, of centrifugal force, and of the 

 disturbing force. The equation to this level-surface is 



-^ 2 s7 2 + G = const ............................ (iii), 



where n is the angular velocity of the rotation, or denotes the distance of any 

 point from the earth's axis, and is the potential of the earth's attraction. 

 If we use square brackets [ ] to distinguish the values of the enclosed quanti- 

 ties at the undisturbed level, and denote by the elevation of the water 

 above this level due to the disturbing potential Q, the above equation is equi- 

 valent to 



(iv), 



approximately, where dfdz is used to indicate a space-differentiation along the 

 normal outwards. The first term is of course constant, and we therefore 

 have 



(v), 



where, as in Art. 206, g=\ -=- (* - ^n 2 ^} (vi). 



Evidently, g denotes the value of ' apparent gravity ' ; it will of course vary 

 more or less with the position of P on the earth's surface. 



It is usual, however, in the theory of the tides, to ignore the slight 

 variations in the value of #, and the effect of the ellipticity of the undisturbed 

 level on the surface- value of Q. Putting, then, r=a, g=yE/a 2 , where E 

 denotes the earth's mass, and a the mean radius of the surface, we have, 

 from (ii) and (v), 



(vii), 



where ffsss 9''' a ........................... (viii) ' 



as in Art. 177. Hence the equilibrium-form of the free surface is a harmonic 

 * Thomson and Tait, Natural Philosophy, Art. 804. 



