64 



On the Tides of the Arctic Seas. 



[Mar. 15, 



In the present discussion I have employed all the hourly observations 

 made during the 23 days, and have obtained the following results : — 



Diurnal Tide. Semidiurnal Tide. 



S = 36'4 inches. S = 5'9 inches. 



i s =3* 2 m . i s =2 h 48 m . 



M = 18'5 inches. M = 15-5 inches. 



V=-2 h 48 m . ^ = 6 h 2| m . 



The present more complete discussion fully confirms the result before 

 obtained by me respecting the great magnitude of the Solar Diurnal 

 Tide at this station, and also shows a satisfactory agreement in the other 

 coefficients obtained from H. W. and L. W. observations only. 



The method employed in the present paper is based on Fourier's 

 Theorem, by which the height of tide is expressed as follows : — 



,F=A 



where 



+Aj cos s + A 2 cos 2s+&c, 



+ B L sin s+B 2 cos 2s + &c, 



F = height of water. 

 s = hour-angle of sun. 



The coefficients A , A 1? A 2 , B v B 2 , &c, being found by well-known 

 formulae, they are again expressed, by Fourier's Theorem, as follows : — 



+ a 1 cos u+a 2 cos 2w+ &c, 



A n =a 



+ b 1 sin ^6 + 6 2 sin 2u+&c, 



where u passes through all its changes in a fortnight, and the coefficients 

 are calculated in a similar manner. 



The known theoretical formulae for the Diurnal and Semidiurnal Tides, 

 expressed in terms of parallax, declination, lunar and solar hour-angles, 

 are now converted into functions of the true and mean anomaly and of 

 the sun's hour-angle, and finally into simple functions of s and u. These 

 expansions are now compared, term by term, with the terms of the tidal 

 expansions found by means of Fourier's Theorem, and the final Lunar 

 and Solar Tidal Coefficients calculated out with ease. 



Although the short period of observation at Port Kennedy (23 days) 

 renders this method of discussion not much more valuable than the usual 

 method of EC. W. and L. W. observations, I have developed it at length 

 in the hope . of applying the method to more complete series of Arctic 

 Tides, which I hope shortly to lay before the Royal Society. 



In developing this method I found it necessary to make use of the 

 following series, for which I am indebted to my friend Mr. Benjamin 

 Williamson, F.T.C.D. :— 



