284 Theory of the Tides 



termination of the coefficient of the series. For these a separate procedure 

 of computation is given. The result for the equator can be found in Table 29 

 in the line "semi-diurnal tides, Laplace". It is found at once that the tides 

 at the poles are direct for all depths. They are also direct at the equator, 

 when the depth of the ocean is greater than 26,250 ft (8000 m); the range 

 becomes practically equal to the equilibrium value when the depth becomes 

 larger. Between 29,000 ft (8850 m) and 14,500 ft (4430 m) there must be 

 a critical depth, for which the tide at the equator changes from direct to 

 inverted. Not far below 7250 ft (2210 m) there seems to be soon a second 

 critical value; with smaller depths the tide is again direct. With inverted tide 

 at the equator and direct tide at the poles, there must be one or more pairs 

 of nodal circles y\ = symmetrically situated on both sides of the equator. 

 With h = 7250 ft (2210 m), this nodal line is at a geographical latitude of 

 about ±18°. 

 (b) Hough's Theory 



120 years after the theory of Laplace, Hough (1897, p. 201 ; 1898, p. 139; 

 see also Hillebrand, 1913) has given a further development and an improve- 

 ment of the Laplace theory. He substituted expansions in spherical harmonics 

 for the series of powers of cos & and sin#; these expansions offer the advantage 

 of converging more rapidly, otherwise the computations are similar to those 

 of Laplace, but he succeeds in considering also the mutual attraction of the 

 particles of water. In Table 29, the values of Laplace have been compared 

 with those of Hough, for the long-period tides. The influence of the mutual 

 attraction consists in a decrease of the amplitude, however this decrease is not 

 especially large except in the periods of the free oscillations, which for ocean 

 depths of 58,100 ft (17,700 m) and 7250 ft (2210 m) are reduced to 9h 52min 

 and 18 h 4min respectively (see previous reference). 



The free oscillations of the second and third species (s = 1 and s = 2) have 

 been especially analysed by Hough. Two waves of equal amplitude travelling 

 around the earth in opposite directions (with and against the rotation) will 

 give by their superposition on a non-rotating earth standing oscillations in 

 a form of spherical harmonic functions. On the rotating earth the waves will 

 have different velocities of propagation, so that normal standing oscillations 

 are split up into waves having the character of tesseral (s = 1) and sectorial 

 (s = 2) spheric harmonics travelling westward and eastward. The waves 

 travelling westward are more important, as they are faster than those travel- 

 ling eastward and because they have the same direction as the tidal forces. 

 Table 30 gives the period in sidereal time for these oscillations "of the first 

 class" which are of most importance in relation to the diurnal and semi- 

 diurnal tides, for different depths. At the same time certain steady motions 

 which are possible without change of level, where there is no rotation are 

 converted into long-period oscillations with change of level. The correspond- 

 ing moves are called as "of the second class". The quickest oscillation of the 



