182 Mr. D. D. Heath on the Dynamical Theory of 



or 



at. the equator, 

 whence, finally, 



_ 2mc (n 2 cos 2 d — m 2 ) 

 (3m + ri){C—gc) 



, — 2mc . m 2 



(Sm + n){C-gc)' 



We see, therefore, that when m < n, not only do Wj and % 



become infinite where cos 0= — , but the expression for the depth 



of the sea becomes negative beyond that latitude ; so that we 

 must suppose a wave of the assumed form to be altogether out 

 of the question in nature, at least without friction, unless the 

 truth be, as suggested above (9), that we ought to take account 

 of vertical effective forces. 



Laplace, and Airy following him, simplify their analysis by 

 making m=n. We then have /e = Zsin 2 0, or the sea thinning 

 out to nothing towards the pole; where u v v l} and also the 

 linear velocity v Y sin 0, all become infinite. This is, as we have 

 seen, so far as the method is trustworthy, the limiting case be- 

 tween possibilities and impossibilities. 



2m 3 

 Generally L, or the value of I when C = 0, is r^ — - — r-. "When 



J ' {3m + n)g 



7l 2 



m=n, this becomes ^-; and then (K) c may be written 

 2C I 



^ n l-2l 

 9 

 which is Airy's form. 



Although the other term of the force B sin 26 cos w cannot 

 subsist by itself, we may inquire with what law of depth the 

 corresponding diurnal wave will have a corresponding form in 

 all latitudes ; and if this law includes the form 



we might, where m>n i have both waves of that character coex- 

 isting ; otherwise the semidiurnal wave will be of an unknown, 

 or at best approximately known form. 



The general remarks as to the connexion of equatorial depth 

 and position of wave hold equally here as in the other case — 

 though the only case we shall treat of forms one of the exceptions. 



