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



where cr is nearly equal to 2n. 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 <r = 2n, 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 



h = h sitf0 ........................... (2)f. 



Adopting an exponential factor e 1 {2nt+ * <a+e] , and putting therefore 

 /= 1, s = 2, we find that if we assume 



'=(7sin 2 ........................... (3) 



the equations (7) of Art. 208 give 



n , * n 



u= Ccote, =- C . - ............ (4), 



m 2m sin 6 



whence, substituting in Art. 208 (4), 



f=2/* /ma.C'sin 2 ..................... (5). 



Putting f= " + f, and substituting from (1) and (3), we find 



2h /ma /t ^ 



and therefore = - , -- ^ -. f ........................ (7). 



* 



For such depths as actually occur in the ocean we have 2h Q < ma, 

 and the tide is therefore inverted. 



It may be noticed that the formula? (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, y=l, 

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



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

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

 t Cf. Airy and Darwin, II. cc. 



