in Laplace's Theory of the Tides. 2.37 



of a diminishes ; but this diminution has a limit, namely the 

 equilibrium value, which it soon approximately reaches. To find 

 what is meant by " soon M (" bientot") take the case of e = \, or 



depth r ,,, r of tne radius. For a rough approximation to R 3 



7/i"<cO 



take R 4 =0, and use formula (9) § 8 with & = 3. Thus we have 



Then by successive applications of (8) with # = 2 and k = l we 

 find 



R 2 = -0367 and R, = - 104. 



Hence in this case we have roughly 



a=H(^ + -104.,r 4 + -104.. -0367. x 6 + '104. -0366. -0185a 8 ) 



= H(a?* + \L04 . a* + -00382 . a 6 + -000071 . x 8 ), 



which shows that when the depth is about a seventieth of the 

 radius, the actual amount of equatorial tide exceeds the equili- 

 brium amount by nearly eleven per cent. 



16. From the first and second of Laplace's numerical formulae 

 for a given above (§ 15), we may infer that when e is increased 

 from 2' 5 continuously to 10, the value of a for any value of x 

 must increase continuously to -f oo , then suddenly become 

 -co, and increase continuously from that till it has the value 

 given by the formula for e= 10. When e has a value exceeding 

 by however small a difference the value which makes a= + co , 

 the value of a for very small values of x is positive, and dimi- 

 nishes through to very large negative values as x is in- 

 creased to 1 ; that is to say, there are nodes coinciding with 

 two very small circles of latitude, one round each pole, direct 

 tides within these circles, and very great inverted tides round 

 the rest of the earth. As e is increased continuously from this 

 first critical value, the nodal circles expand until (as seen above) 

 when e=10 they coincide approximately with 18° North and 

 South latitude. From the greatness of the coefficient of x 4 

 in Laplace's result for this case we may judge that e cannot be 

 increased much above 10 without reaching a second critical 

 value, for which the coefficient of x 4 , after increasing to 

 -f oo , suddenly becomes — go . It is probable that the nodal 

 circles do not get much nearer the equator than 18° North and 

 South before this critical value is reached. When e is increased 

 above it, a second pair of nodal circles commence at the two 

 poles, spreading outwards and getting nearer to the former pair 

 of nodal circles, which themselves are getting nearer and nearer 

 to the equator. Then there are direct tides in the equatorial 

 belt, inverted tides in the zones between the nodal parallels of 



