November 11, 1898.] 



SCIENCE. 



645 



ing the results, aud the short waves, where 

 this vertical acceleration may not be ne- 

 glected in the equations of motion. I shall, 

 as far as possible, deal with them in this 

 order. 



The neglect of the vertical acceleration, 

 which implies that the dynamical pressure 

 is equal to the statical pressure, greatly 

 simplifies the problem of wave-motion. 

 Neglecting compressibility and viscosity, 

 the ordinary problems of long wave-motion 

 are not very difficult. The motion of long 

 waves in canals is an old problem and was 

 used by Airy for explaining the tides. A 

 paper by McCowan in 1892 on these long 

 waves when the section of the canal is uni- 

 form must be mentioned. Usually the 

 canal has been taken to be of rectangular 

 section. McCowan treats one which has 

 sloping sides. Incidentally he obtains a 

 canal of such shape that these long free 

 waves are propagated along it without 

 change of form. But a more important 

 result is the detection of a serious error in 

 Airy's explanation of the double high-tide, 

 sometimes observed in estuaries and rivers. 

 The method adopted was, as usual, one of 

 continued approximation, and Airy fell into 

 the error — not unknown to mathematical 

 physicists — that of carrying his approxima- 

 tions further than the initial assumptions 

 warranted. It appeared that the wave 

 might divide into two or even three waves. 

 McCowan shows that this division of the 

 wave is without foundation when the equa- 

 tions are correctly treated and that, there- 

 fore, Airy's explanation of the double high- 

 tide fails. The double high-tide is still 

 unexplained. 



The general theory of tides is fully dealt 

 with and brought up to date by Professor 

 Darwin in his article in the Encyclopaedia 

 Britannica of date 1888. Two important 

 papers have appeared since then. To re- 

 port how these two papers have advanced 

 our knowledge of tidal theory, a few re- 



marks must be made on the general prob- 

 lem. 



The theory of the tides is mainly one of 

 forced oscillations. The sun and moon, 

 moving in orbits round the earth whose 

 essential nature is periodic, their motions 

 are expressible by means of sums of sines 

 or cosines of angles which vary with the 

 time, and the periods of these terms diifer. 

 The difference of the attractions on the 

 earth and the water which covers it pro- 

 duces oscillations with periods which cor- 

 respond to those of the sun and moon. 

 Hence the periods of the principal tides 

 will be known in advance. ISTotwithstand- 

 ing this fact, the general equations which 

 express the motion of the water have 

 hitherto only been integrated on the fol- 

 lowing assumptions : First, that the parti- 

 cles of water never move far from their 

 mean position in comparison with the 

 radius of the earth — an assumption which 

 is easily justified and need not be discussed. 

 Secondly, that the ocean covers the whole 

 earth supposed spherical. Thirdly, that 

 the depth of the ocean is uniform or a func- 

 tion of the latitude only. Fourthly, that 

 the attraction of the ocean on itself is to 

 be neglected. The last phrase simply 

 means that we neglect the difference of 

 the attraction of the water on itself in its 

 actual form from its attraction in the 

 nearly spherical form it would assume if 

 there were no disturbance. 



The irregular shapes of the bed and 

 shores of the ocean make calculation of the 

 effects due to them almost impossible. But 

 the attraction of the ocean on itself should 

 be susceptible to calculation. Hitherto 

 this has always been neglected, owing to 

 mathematical difficulties. Poincare and 

 Hough, in two papers which have latelj' 

 been published quite independently and 

 nearly at the same time, have taken it into 

 account. Of these two papers Hough's is the 

 most important in view of the applications. 



