56 REPORT — 1886. 



raodificatioiis may be introduced. I spared, however, no pains to reduce 

 the labour of computation. Nearly half the work in forming a short 

 tide-table is preparatory, and would serve for a systematic computation of 

 tables for all time. 



III. An Attempt to Detect the IQ-Yeaely Tide. 



If M, E be the moon's and earth's masses ; a the earth's mean radius ; 

 c the moon's mean distance ; u the obliquity of the ecliptic ; i the inclina- 

 tion of the lunar orbit ; e the eccentricity of the lunar orbit ; S the 

 longitude of the moon's node ; and X the latitude of the port of observa- 

 tion ; then the term in the equilibrium tidal theory which is independent 

 of the moon's longitude (see Schedule B, iii., Report of 1883) is 



_ _j j - j a Os — I sin^ \) (l-}-| e^) sin i cos i sin w cos w 



^JiiycJ - [-cos£3-|-:^tanitanwcos2 £3]. 



Since | tan itan w= "00975, the second term is negligeable compared with 

 the first. 



If we take 



^=-l_, -= J^^„, a=21 xlO« feet, 'i=5° 8', a>=23° 28', 

 J? 81-5' c 60-27' 



the expression for this tide is, in British feet, 



—0-0579 (i-f sin^ X) cos S . 



Thus, at the poles this tide gives an oscillation of sea-level of 0-695 of 

 an inch, or a total range of 1 1 of an inch, and at the equator it is half as 

 great. 



In the ' Mecanique Celeste ' Laplace argues that all the tides of long 

 period (such as the fortnightly tide) must conform nearly to the equili- 

 brium law. I shall adduce arguments elsewhere ' which seem to invalidate 

 his conclusion, and to show that in these tides inertia still plays the 

 principal part, so that the oscillations must take place nearly as though 

 the sea were a frictionless fluid. 



With a tide, however, of as long a period aa nineteen years Laplace's 

 argument must hold good, and hence the equilibrium tide of which the 

 above is the expression must represent an actual oscillation of sea-level, 

 provided that the earth is absolutely rigid. The actual observation of the 

 19-yearly tide would therefore be a result of the greatest interest for 

 determining the elasticity of the earth's mass. 



A reduction of the observed tides of long period at a number of ports 

 was carried out in Thomson and Tait's ' Natural Philosophy,' Part II., 

 1883, in the belief in the soundness of Laplace's argument with regard to 

 those tides, and the conclusion was drawn that the earth must have an 

 effective rigidity about as great as that of steel. The failure of Laplace's 

 argument, however, condemns this conclusion, and precludes us from 

 making any numerical conclusions with regard to the rigidity of the 

 earth's mass, excepting by means of the 19-yearly tide. The results 



' In an article on the Tides in the Eneyclopadia Britannica. The section of tlie 

 article ' On the Tides of Long Period ' will probably be communicated also to tlie 

 Eoyal Society. 



