Theory of the Tides 287 



1928, p. 692) has also treated the case of an ocean extending from pole to pole 

 and bordered by two meridians (bi-angle on the sphere). The mathematical 

 difficulties become greater and require working with an infinite number of 

 functions, in order to meet the condition that, for the semi-diurnal tides, the 

 velocity perpendicular to the equator be zero, same as at the shores of the ocean. 

 Goldsborough has further given a solution for the case that the depth varies 

 with the square of the cosine of the latitude and is bordered by two meridians 

 which are 60° apart. The mean depth has been selected at 15,500 ft (5200 m). 

 The conditions are roughly like those of the Atlantic Ocean. He had found 

 previously that large semi-diurnal tides cannot develop in a polar basin for 

 the given depths, from which he inferred that the Atlantic Ocean cannot get 

 its large semi-diurnal tides from the Polar Ocean through simple propagation; 

 they must be increased through resonance. This seems to be confirmed by 

 the fact that the critical depth in the case under consideration is about 

 16,880 ft (4800 m), whereas the mean depth of the Atlantic Ocean is about 

 12,700 ft (3900 m). The condition of resonance would be fulfilled at this 

 depth by a bi-angle with an opening of 53°. 



Proudman and Doodson (1928, p. 32) have contributed important papers 

 to the problem of tides in bounded oceans. The former has extended Taylor's 

 method, which he used for the computation of the tides in a rectangular 

 basin and which was discussed on p. 210, to ocean basins with a different 

 configuration. He reduced it in a general way to an infinite system of linear 

 equations, which can be solved according to the method of the infinite de- 

 terminants. He thus was successful in deriving the tides of a semi-circular 

 ocean of uniform depth. An application of this solution is given in the dis- 

 cussion of the tides of the Black Sea, which can be approximately compared 

 with a sea of this form. Much more difficult is the computation of the tides 

 of a hemispheric ocean bordered by a complete meridian. Doodson has de- 

 veloped a numerical method of integration of his own, which, however, has 

 not yet been applied in practice. Goldsborough (1931, p. 689; 1933, 

 p. 241) has also dealt with this problem, but his method is rather complicated 

 and there are no numerical results. Proudman (1916, p. 1 ; 1931, p. 294) has 

 recently shown that a solution of an infinite series of simultaneous equations 

 takes care of the problem, but that the solution of a finite number of equations 

 can already give a quite sufficient approximation. The mathematical de- 

 velopments are very difficult and the work in connection with the numerical 

 computation, even for one single depth, is most extensive. Such computations 

 have been made by Doodson (1936, p. 273; 1938, p. 311) for the partial 

 tide K x (a = co) and K 2 {a = 2g>) for various depths. The essential results 

 of these important investigations are given summarily as follows. 



In the following figures the tide is represented by lines of equal phase 

 (co-tidal lines) and by lines of equal amplitude (co-range lines). The numbers 

 on the co-tidal lines represent the phase in degrees. The zero-value is the phase 



