324 BEPORTS ON THE STATE OF SCIENCE. — 1920. 



precise extent the purely astronomical tide at any station may be expressed as a 

 series of a reasonable number of harmonic constituents. When this has been done 

 and the methods of analysis and prediction refined so as to give predictions correct 

 to this extent, a hopeful investigation may be made into the residual astronomical 

 tide and the whole of the meteorological disturbance. 



In a preliminary report * presented to the Geophysical Discussion of June 1918, 

 it was stated that tide tables as at present produced appear to be adequate for 

 practical needs. This was based on the facts that the practically important 

 constituents can be determined fairly accurately, and that harmonic prediction 

 presents no theoretical difficulties like those of harmonic analysis. The investiga- 

 tions of Dr. A. T. Doodson show, however, that the published tables of harmonic 

 predictions are also very unsatisfactory. 



Harmonic Tidal Constituents. 



2. The gravitational forces generating the tides are derivable from a potential 

 wliich is everywhere proportional to what the height of the tide would be if water 

 covered the whole earth and had lost its inertia without losing its gravitational 

 properties. 



Such a tide — the equilibrium tide — may be calculated by adding the amounts by 

 which a certain pair of nearly spherical surfaces of revolution project above the 

 mean water-level. Each of these surfaces encloses a volume equal to that of the 

 earth, and is slightly variable in shape. They move so that their axes, while always 

 passing through the centre of the earth, pass also always through the centres of 

 the sun and moon respectively. 



The tides due to either of these spheroids may be expressed as a series of 

 constituents, each of which varies harmonically in a period determined by 

 astronomical data. From dynamical principles it follows that to each of these 

 constituents there will correspond a similar constituent in the actual tides, that is, 

 a constituent varying harmonically in the same period. 



To find, in the actual tides at any station, the amplitude of each of these 

 constituents, together with the lag of its phase behind that of the corresponding 

 constituent of the generating potential, is the object of the harmonic analysis of 

 tidal observations. 



Let us consider the speeds of the constituents of lunar origin ; we have to 

 examine the motion, relative to any point on the earth's surface, of the spheroid 

 whose axis passes always through the moon. 



The pole of this spheroid which is nearer the moon is a little further from the 

 earth's centre than is the opposite pole, while the whole departure from sphericity 

 depends on the distance of the moon. 



Let 7 denote the angular speed of the earth's rotation and (t the mean motion of 

 the moon. 



If the moon moved with constant angular speed in the plane of the equator and 

 at a constant distance from the earth, we should have, at any station, high water 

 occurring regularly at intervals of w/(7 — o-), with a maximum range of tide at the 

 equator. The rise and fall of the water would not quite be simply harmonic, but 

 could be resolved, with sufficient accuracy, into a harmonic constituent of speed 



2(7 - <r), 



of amplitude inversely proportional to the cube of the moon*s distance, and two 

 much smaller constituents of speeds 



7 - cr, 3(7 - <r) 



and of amplitudes inversely proportional to the fourth power of the moon's distance. 

 The fact that the moon does not move as here supposed causes many modifications, 

 but it is only on the constituent of speed 2(7 — tr) that their effect need be 

 considered. 



Let us still suppose the moon to move in the equator, but take into account the 



* Brit. Assoc Jteport for 1918, pp. 16, 16. 



