On the " Oscillations" in Laplace's Theory of Tides, 279 



where K 4 is absolutely indeterminate as regards the general 

 solution of the original equation (the only solution of which La- 

 place is treating here), and where of course we may put K 4 or 



K 4 



— as may be convenient. And in Article 113 I have pointed out 



that the value of K 4 , undetermined in the general solution of 

 the original equation, may be so determined as to apply to a sea 

 bounded on its north or south side. And this is the legitimate 

 use of such an undetermined constant. 



G. B. Airy. 

 Roval Observatory, Greenwich, 

 September 11, 1875. 



XXXI. Note on the " Oscillations of the First Species" in La- 

 place's Theory of the Tides. By Sir William Thomson, 

 F.R.S.* 



LAPLACE'S " Oscillations of the First Species" are simple 

 harmonic oscillations, in which the surface of the water is 

 always a figure of revolution round the axis of rotation. The 

 '•tide-generating influence'''' in this case is such that the equili- 

 brium tide-height would be © cos at at time t, a denoting a con- 

 stant (called the " speed " in the British- Association Tidal Com- 

 mittee's Report for 1871), and © a function of the latitude. 

 @ being supposed known, the problem consists in finding a' a 

 function of the latitude such that (® + a') cos at is the actual 

 tide-height at time t, and, for the case of the sea equally deep 

 everywhere, is to be solved by finding the proper solution of the 

 differential equation 



A. (\zj£ <M\ _ w-4,0 . m 



dp\jj*-f*dp) * ea ~* eU > • • • • v 1 ) 



where fi denotes the sine of the latitude, and e and / are con- 

 stants defined by the equations 



a n^r 2 mr 



r being the earth's radius, 



g the force of gravity at its surface, 



m the ratio of gravity to equatorial centrifugal force, being 



equal to ^ J-g, 

 n the angular velocity of the earth's rotation, 

 and 7 the depth of the sea, supposed small in comparison 



with r — not greater, say, than ^W. 

 The quickest of the " Oscillations of the First Species " is the 



* Communicated bv the Author. 



