TIDAL OBSERVATIONS, 149 



72. The fact noticed, that one of the variation-tides has the same period 

 as the solar semidiurnal tide, would be of great importance for tidal theory 

 were it not that its magnitude must be so small as to be scarcely sensible. 

 According to the values of the perturbations in longitude and radius vector, 

 due to variation shown in Hansen's Tables, the equilibrium value of the solar 

 semidiurnal variation-tide would be only about -^ of the equihbrium value 

 of the 2(y—2a + r]) variation-tide. The dynamical value of the latter tide, 

 shown in § 61 above, is only about a quarter of a foot above and below the 

 mean level. Hence it cannot be expected that the smaller component should 

 sensibly influence the observed results obtained by the analysis which had 

 been undertaken for the solar semidiurnal tide. The close agreement between 

 the results, both for amplitude and for epoch, of the chief component varia- 

 tion-tide, 2(y — 2(7 -f- J?), for the four years, is so satisfactory that the evalua- 

 tion is undoubtedly genuine in this case. 



73. The same may be said also of the Helmholtz luni-solar tide. The 

 effect of this tide is of great practical importance, especially in respect to 

 navigation. It is well known that, generally in harbours, estuaries, and 

 channels, the tide rises faster than it falls. The harmonic analysis of this 

 phenomenon was given first, I believe, by Airy: see for example, his formula 

 for Deptford in his article " Tides and Waves," Encyclopaedia Metropolitan a, 

 Helmholtz's admii-able explanation of the grave and acute notes heard ac- 

 companying two loud musical notes sounding simultaneously, referred to in 

 § 24 of the Committee's First Eeport, contains a perfectly general statement 

 of an extremely simple character, which, mutatis ynutandis, is applicable to 

 the tides in every case in which the range of rise and fall is sensible in com- 

 parison with the mean depth through any considerable area of sea or channel 

 influencing them. The application of this theory, for example, to a tide of 

 10 feet whole range from mean level, considered as the resultant of two 

 simultaneous tides of 5-feet range, suggests immediately the harmonic terms 

 (such as those calculated by Aiiy for Deptford) which express the phenomenon 

 of the rise being more rapid than the fall. The same theory applied to the 

 spring-tides (which are the resultant of the lunar and solar semidiurnal tides 

 M'hen they agree in phase) and to the neap-tides (which are the resultant of 

 the same components when they are opposite in phase), shows that the terms 

 expressing deviation from the simple sum of the two chief harmonic terms must 

 be greater than the sum of the deviations about the time of spring-tides, and 

 less than the algebraic sum of the contrary deviations about the time of the 

 neap-tides. Helmholtz's general statement again suggests instantly the 

 harmonic terms proper to express the anticipated result. In this case it is 

 the second of the two " shaUow-water tides " indicated in § 24. The deter- 

 mination of this term from observation, promised at Norwich, 1868, as an 

 early undertaking of the Committee, has now been accomplished, both for 

 Liverpool and Eamsgate, by Mr. Roberts, in consequence of his having for 

 some time been baifled by discrepancies in his investigation of the Liverpool 

 tides showing an approximately quarter-diurnal period when all the com- 

 ponents previoiisly evaluated were properly put together to express the tides 

 on the days selected for the comparison between theory and observation 

 described above. The period of this new component is a quarter of the har- 

 monic mean of the mean solar and mean lunar days. Its amplitude (that is, 

 half the range from lowest to highest) is about f of a foot at Liverpool and 

 g of a foot at Eamsgate. Both it and the previously found shallow-water 

 components are greater in proportion to the chief tides at Eamsgate than at 

 Liverpool ; this no doubt is due to the great extent and shallowness of the 



