212-214] DIURNAL AND SEMI-DIURNAL TIDES. 357 



where a is nearly equal to 2?i. This includes the most important of 

 all the tidal oscillations, viz. the lunar and solar semi-diurnal tides. 



If the orbital motion of the disturbing body were infinitely 

 slow we should have a = 2ra, and therefore /= 1 ; for simplicity 

 we follow Laplace in making this approximation, although it is 

 a somewhat rough one in the case of the principal lunar tide*. 



A solution similar to that of the preceding Art. can be obtained 

 for the special law of depth 



li = h Q $itfd ........................... (2)f. 



Adopting an exponential factor e* lmt+2m+e} , and putting therefore 

 f= i t s = 2, we find that if we assume 



? =asin 2 ........................... (3) 



the equations (7) of Art. 208 give 



. ~ ... 



u= Ocot 6, 0=- C . a ............ (4), 



m 2m sm 6 



whence, substituting in Art. 208 (4), 



= 2^0/ma. (7sm 2 ..................... (5). 



Putting f = &quot; + and substituting from (1) and (3), we find 



,_ 



and therefore b = ~ n - rn ? ........................ ( ) 



1 - 2/io/ma * 



For such depths as actually occur in the ocean we have 2/& &amp;lt; ma, 

 and the tide is therefore inverted. 



It may be noticed that the formulae (4) make the velocity 

 infinite at the poles. 



214. For any other law of depth a solution can only be 

 obtained in the form of an infinite series. 



In the case of uniform depth we find, putting s = 2, /=!, 

 4raa//i = in Art. 208 (8), 



(1 - /* a y |J + {/3 (1 - ^ - 2p* - 6} ? - - /3 (1 - ^) 2 ? p -(8), 



* There is, however, a luni-solar semi-diurnal tide whose speed is exactly 2/i 

 if we neglect the changes in the planes of the orbits. Cf. p. 355, footnote. 

 t Cf. Airy and Darwin, II. cc. 



