646 



SCIENCE. 



[N. S. Vol. VIII. No. 203. 



Previous writers, and particularly Darwin, 

 have used simple sums of harmonic terms. 

 Hough uses Zonal Harmonics and finds 

 that this enables him to include the attrac- 

 tion of the ocean on itself. In particular 

 the fortnightly tide has to be altered from 

 5 to 8 per cent, owing to this cause. Poin- 

 care makes a rough estimate only, which is 

 double this amount. At the same time he 

 points out that the irregular shape of the 

 continents may alter the coefiicients of cer- 

 tain terms to a large extent. In conse- 

 quence of this he criticises the arguments 

 of Thomson and Tait, who attempt to ex- 

 plain the difference between theory and 

 observation by the solid tides which the 

 moon and sun must produce in a solid 

 earth possessing slight elasticity. He leaves 

 the question open, however. Poincare's 

 paper contains much more in the way of 

 theoretical researches, which must be 

 omitted here. Hough also determines the 

 free oscillations by means of an indefinite 

 determinant. 



Wben we do not neglect the vertical ac- 

 celeration of the particles of fluid the 

 mathematical treatment alters and becomes 

 more difficult. It is in general supposed 

 that the height of the wave is small in 

 comparison with the distance between the 

 crests. In 1883 Lord Eayleigh discussed 

 the short waves which are seen in front of 

 an object trailed along the water, and also 

 the longer waves that are left behind. His 

 method is to find the elfect produced by a 

 line of disturbance inclined at an angle to 

 the direction of steady motion of the fluid. 

 The effects observed by Fronde in his 

 famous experiments on ship-waves are 

 fairly well accounted for. This year a 

 j)aper by Mitchell has just appeared in 

 which he calculates the wave- resistance due 

 to a body shaped approximately like a ship. 

 Again the results are not unsatisfactory. 

 Xord Kelvin has solved a similar problem in 

 several articles in the Philosophical Magazine 



for 1886-7. He considers the effect of small 

 inequalities in the bed of a stream. Of 

 special interest are the standing waves pro- 

 duced by a stone, or by a hole in the stream- 

 bed, or by a wavy bottom, such as we some- 

 times see left on the sea shore by the 

 retreating tide. 



When the wave-height, wave-length and 

 the depth of the fluid are comparable in 

 magnitude the problem becomes very diffi- 

 cult, as the waves are not in general propa- 

 gated without change of shape. They were 

 first treated by Stokes. In 1889 von Helm- 

 holtz took up the general problem and con- 

 sidered the efiect of the wind in making 

 permanent waves. Mitchell has traced the 

 free-wave in the case of infinite depth and 

 its changes as it proceeds. Incidentally he 

 finds again the case where the crest be- 

 comes a cusp with an angle of 120°, a result 

 predicted by Stokes from simple considera- 

 tions. A paper by W. "Wien should be 

 mentioned ; in it the forms of the waves at 

 the surface of separation of the two media 

 are discussed, and forms for the waves pro- 

 duced by the winds, more or less closely 

 approximating to actual phenomena, are 

 found. The wave-fronts are figured by 

 means of Schwarz's method of representing 

 conformably a lemniscate on a circle. 



The solitary wave has received a fair 

 share of attention . In this wave the height 

 is not necessarily small compared with the 

 depth of the fluid, and it may travel for a 

 long distance along a uniform canal with 

 little or no change of form. Experiments 

 have been made and the results published 

 by Scott Russell. Boussinesq and Lord 

 Rayleigh have lately investigated his figures 

 from the mathematical standpoint, and have 

 given a simple formula for the relation be- 

 tween the wave-length and the height which 

 agrees with that deduced by Scott Russell 

 from experiment. Korteweg and de Vries, 

 in 1895, extended the theory of the solitary 

 wave and showed that a new type — named 



