servations at frequent intervals, especially of currents below the surface, are usually made by current meters. One of the commonest of these the Ekman meter registers the mean speed and direc- tion of the current during the interval of time it is in operation, the former by a small propeller actuating a revolution counting appa- ratus and the latter by a vane attached to an apparatus dropping shot into sectorial boxes on a compass card. It has thus to be hauled up to the surface for each reading. Continuous recording instruments are much needed and though some have been invented they do not appear to have been much used. Owing chiefly to the trouble of keeping a meter fixed relatively to the bottom, the accu- rate measurement of currents is a matter of great difficulty. Other data for residual currents or drifts are given by observations of weighted bottles or other forms of floating bodies, or by instru- ments so contrived as to float near the sea bottom.
General Distribution of Off-shore Tides. Much attention is now paid to the " amphidromic points," at which there is no rise and fall of the water and out from which the co-tidal lines radiate. Harris' charts of co-tidal lines contain a number of these points and so does the new chart of R. Sterneck (Sitzb. d. Akad. Wissensch., Wien, 129, 1920), which is based on all available data.
All recent charts of co-tidal lines for the North Sea agree in placing an amphidromic point in the southern region, and one of the serv- ices of the Fav6 gauge has been to give fresh observational veri- fication of its existence (Comptes Rendus, 151, p. 803, 1910).
Dynamical Theory of the Tides. As regards the tidal dynamics of completely defined bodies of water, the only basins which had yielded to mathematical treatment up to 1914 were those of a flat circular sea, the depth of which was a function only of the distance from the centre, and an ocean covering the whole globe with the depth a function only of the latitude. The details for zonal basins of uniform depth have since been worked out by G. R. Goldsbrough (Proc. London Math. Soc., 14, 1914; 15, 1915).
Two attempts have been made, however, to bring some of the latest results of pure mathematics to bear on the general problem. In 1910 Poincard published his transformation of the dynamical equations from the differential to the integral form (Lemons de Mecanique Celeste, t. 3). The theory of integral equations has grown up almost entirely since 1900; its results are perfectly general and are stated explicitly in terms of direct operations. But in the case of tidal problems the arithmetical labour necessary to carry out these operations is so prodigious as to prove quite prohibitive even for the reproduction of known solutions: nevertheless, the theory is valu- able for the establishment of existences.
Utilizing these existence-theorems J. Proudman (Proc. London Math. Soc., 18, 1917) has been able to specify the tidal state of an ocean by means of an infinite number of coordinates of the Lagran- gian type, and then to transform the differential equations into an infinite set of linear algebraic equations. This has afforded a real prospect that the number of geometrically simple basins for which the tidal dynamics is completely known, may be increased.
The explanation, on dynamical principles, of the observed fea- tures of tides in small seas has been considerably advanced, chiefly by A. Defant and R. Sterneck. See Denkschr. d. Akad. Wissensch., Wien, 96 (1919). Sitzungsberichte, 123 (1914), 124 (1915), 129 (1902). The method of treatment only applies to elongated bodies of water and applications have been made to the Red Sea, the Persian Gulf, the English Channel, the Irish Sea and the Adriatic Sea. The motion is assumed to consist of a longitudinal oscillation sustained chiefly by the tides outside, with a transverse surface gradient sustained by the longitudinal current through the earth's rotation.
Other parts of the dynamical theory which have undergone development are those relating to slowly rotating seas and oceans, limiting forms of long period tides and the diffraction of tidal waves. See Rayleigh, Proc. Roy. Soc. (A) 82 (1909); J. Proudman, Proc. London Math. Soc., 12 (1913), 13 (1913), 14 (1914)- In tms con - nexion it may be mentioned that there is an erroneous statement in 26 -957 34, to the effect that the existence in the ocean of continen- tal barriers would have the same effect as that attributed by Laplace to friction. In the actual oceans limiting forms of long period tides are possible which do not take the " equilibrium " values.
Harmonic Analysis. From 1883 up to the present time the stand- ard harmonic development of the generating potential has been that of G. H. Darwin. Quite recently A. T. Doodson .has made a new development, working to a much higher order of approximation than Darwin, and has found that there is a very large number of other constituents which, while certainly being smaller than those of Darwin, are not very much smaller, and in their aggregate may be important. In other words, the convergence of the series of con- stituents is not so rapid as has been assumed.
A corresponding state of affairs exists with regard to over tides and compound tides. For certain British stations A. T. Doodson, being led by dynamical principles, has found it possible to obtain a practically complete representation of the quarter diurnal tides, but it involves many more harmonic constituents than have ever been sought for by the customary methods. This representation is sus- ceptible of very simple algebraic statement and numerical applica- tion but cannot be used on the existing predicting machines.
The present state of analysis is not satisfactory. The harmonic constants do not represent completely the records analysed: for
certain British stations the discrepancy may have a quarter-diurnal range of one foot and a semidiurnal range of one foot.
J. Proudman (British Assoc. Report, 1920, p. 323) has given an account of British work on harmonic analysis with a bibliography and lists of analyses made.
Tide Tables cannot be regarded as satisfactory even for such prac- tical purposes as docking large vessels or navigating over shallows, while for a hopeful study of meteorological effects they are almost useless. The main deficiency appears to be one of analysis of records ; for others, see A. T. Doodson, Brit. Assoc. Report, 1920, p. 321.
When the astronomical tides can be predicted with the same degree of accuracy as the resultant tides can be observed, there appears to be no reason why short date predictions of meteorological tides obviously of great importance should not be attempted.
Atmospheric Pressure and Wind. The effects of meteorological influences on the tides have been much studied, especially by the Scandinavians. As regards the relative importance of atmospheric pressure and wind, a general conclusion appears to be that at a station in the immediate neighbourhood of a wide expanse of deep ocean, the direct pressure effect predominates, whereas at a station in a landlocked and shallow sea, the wind effect predominates. The detailed study of these effects is rendered very difficult by the un- certainties in the predictions of astronomical tides, and most inves- tigations have dealt with mean effects over long intervals of time. There is much literature on the subject : see for example D. la Cour, Danske Meteorologiske Institut, Meddelelser, I (1913), 4 (1917); R- Witting, Fennia 39, 5 (Helsingfors 1918). The 1917 memoir of la Cour is a detailed study of the effects of a storm.
Friction. If tidal motion were everywhere non-turbulent then the amount of friction in the oceans would be quite insufficient to account for the outstanding discrepancy between theory and obser- vation in the motion of the moon. See R. O. Street, Proc. Roy. Soc. (A) 93 (1917). But the motion associated with large tides in shallow seas is undoubtedly turbulent and though this has long been rec- ognized it is only recently that numerical estimates of its amount have been made. See G. I. Taylor, Phil. Trans. (A), 220 (1920). H. Jeffreys, ibid 221 (1920), concludes that the total amount of friction is just about sufficient to account for the discrepancy men- tioned. He takes the chief contributing areas to be Bering Sea, the Yellow Sea, Malacca Strait and the American N.W. Passage.
History and Bibliography. To the list of outstanding names in the history of the theory of the tides should be added those of G. H. Darwin and H. Lamb. The chief contributions of the former were his elaboration of the methods of harmonic analysis and his far- reaching cosmogonical deductions as to the consequences of tidal friction. The chief contribution of the latter is in connexion with steady motions and the discrimination of free oscillations in the general dynamical theory. Additions to the list of books on tides are R. A. Harris, Manual of Tides v. (1907); H. Poincare, Legons de Mecanique Celeste, t. 3 (1910) and O. Kriimmel, Handbuch der Ozeano- graphie, B. 2, C. 3 foil). (J. P.*)
TIENTSIN, China (see 26.963). After the Chinese revolution of 1911 the political and social importance of Tientsin considerably increased, inasmuch as it became an unofficial place of residence, and often of refuge, for high Chinese officials in times of trouble, and a neutral ground convenient for the conferences' of the northern military governors. The number of Chinese residents in the Foreign Concessions steadily increased after 1912 and building operations continued unabated. The World
War naturally led to a greatly increased demand in most branches of the port's export trade, so that, in spite of floods, famines and brigandage, business was very prosperous after 1915- Local industrial enterprise was stimulated by the curtailment of imports from Europe and a lack of shipping facilities. In 1919 it received a fresh impetus from the boycott of Japanese goods,
with the result that many new factories were established for the manufacture of goods heretofore imported.
The city produces cotton yarn in steadily increasing quantities, seed and groundnut oils, canvas, leather, soap, candles and numerous articles for domestic consumption; its chief exports during the war were furs and skins, wool, bristles, strawbraid, carpets and prepared eggs. The value of the export trade in 1919 was 71 million taels, as against 51 millions in 1918, and 42 millions in 1917. Tientsin's prosperous growth was indicated by the opening of four new Chinese banks in 1919. The coal trade, from the Kailan Mines in Chihli and the Peking Syndicates in Honan, has greatly increased since the adoption of the system of Anglo-Chinese cooperative working.
In Sept. 1917 the Foreign Concessions and the trade of the port suffered severely from floods, which burst the banks of the Grand Canal and inundated all the plain surrounding the city. Before normal conditions could be restored in the British, French and Japanese Concessions, the municipal councils were compelled to surround them with dikes and pump out the water. This flood also produced a shoaling of the river bar at Taku, with results seriously prejudicial to the trade of the port. With a view to preventive and remedial
