Theory of Long Waves. 265 



at which the equations indicate discontinuity, a point far 

 beyond that permissible under the fundamental assumptions. 

 It is clear, therefore, that the subdivision obtained by Airy 

 must be due to his having extended his interpretation of his 

 approximation to a distance at which it ceased to give a fair 

 account of the motion. In fact, returning to (34), if we put 

 n/s/gk=tn, take q — and k=2\/gh, we shall get at once 

 for an approximation, 



77=770 sin {nt — m!; — %m£.r)]h), 

 or 



17 = 770 sin (nt — mf) — } .r) 2 /h ,m£ sin 2(nt— wif); . (46) 



which is the solution used by Airy. He has given a tracing 

 of the wave-form represented by this equation, which shows a 

 subdivision of the wave at a sufficiently advanced station. 

 This tracing, with a brief summary of Airy's results, is repro- 

 duced by G. H. Darwin in his article " On Tides," in the 

 Encyclopaedia Britdnnica. Airy has further carried (46) to a 

 third approximation, which may easily be obtained from (34) 

 and need not be reproduced here, and from this he finds there 

 may even be a subdivision into three. All this, however, 

 follows inevitably from the form of approximation used, that 

 of a series of sines of multiple arcs, when due regard is not 

 paid to the limits within which only it can be regarded as an 

 approximation at all. In fact such an approximation will 

 indicate a subdivision into as many waves as it contains terms; 

 but the more terms there are taken, the further up the river 

 must we advance before any subdivision is indicated. That 

 great caution is necessary in applying such a series to points 

 far up the river is further indicated by the occurrence of 

 powers of £ in the coefficients. 



It would be equally simple to discuss the tidal phenomena 

 produced in a river by the oceanic tides given by (33) ; but 

 practically, the component tides having nearly equal periods, 

 the discussion of the simpler tide, given by (32), suffices to 

 give a fairly accurate account of the more complex, if we 

 regard the range as having a slow periodic variation, the 

 period being the interval between successive spring tides ; 

 and in fact, in referring to the effects of high and low tides, 

 this variation was kept in view in the preceding discussion. 



Dundee, December 21,. 1891. 



