362 TIDAL WAVES. [CHAP. VIII 



be very small, and the formulae (17) and (19) then shew that the 

 tide has everywhere sensibly the equilibrium value, all the coeffi- 

 cients being small except the first, which is unity. As h is 

 diminished, /3 increases, and the formula (17) shews that each 

 of the ratios Nj will continually increase, except when it changes 

 sign from + to by passing through the value oo . No singu- 

 larity in the solution attends this passage of Nj through oo , 

 except in the case of NI, since, as is easily seen, the product 

 Nj^Nj remains finite, and the coefficients in (19) are therefore all 

 finite. But when NI = CO , the expression for becomes infinite, 

 shewing that the depth has then one of the critical values 

 already referred to. 



The table above given indicates that for depths of 29040 feet, 

 and upwards, the tides are everywhere direct, but that there is 

 some critical depth between 29040 feet and 14520 feet, for which 

 the tide at the equator changes from direct to inverted. The 

 largeness of the second coefficient in the case /3 = 40 indicates that 

 the depth could not be reduced much below 7260 feet before 

 reaching a second critical value. 



Whenever the equatorial tide is inverted, there must be one 

 or more pairs of nodal circles (f = 0), symmetrically situated on 

 opposite sides of the equator. In the case of /3 = 40, the position 

 of the nodal circles is given by v = '9o, or = 90 18, approxi- 

 mately *. 



215. We close this chapter with a brief notice of the question 

 of the stability of the ocean, in the case of rotation. 



It has been shewn in Art. 197 that the condition of secular 

 stability is that V T should be a minimum in the equilibrium 

 configuration. If we neglect the mutual attraction of the elevated 

 water, the application to the present problem is very simple. The 

 excess of the quantity V T over its undisturbed value is evidently 



s (i), 



where "^ denotes the potential of the earth's attraction, &S is an 

 element of the oceanic surface, and the rest of the notation is as 



* For a fuller discussion of these points reference may be made to the original 

 investigation of Laplace, and to Lord Kelvin's papers. 



