of Laplace's Differential Equation of the Tides, 399 



cos (at + syjr), 



?= ST", ' 7*—^ £ • cos (^ + ^)> 55; 



4wz(l— /a 2 ) « 



1 . J> . 1 rf(£ 



where / denotes — , and m the ratio of equatorial centrifugal force 



to gravity, 



This fully determined solution expresses the motion of the 

 supposed zonal ocean due to a disturbing influence, of which 

 the equilibrium-tide height is E cos (at -{-sty), E being expressed 

 by the formula 



E = (w + n 1 /a + n a ^ + &c), . . (56) 



ns 



where n , n v w 2 , &c. are any given numbers. 



13. If L = 0, equation (54) gives, except in a certain critical 

 case to be considered presently, C = and K =0, and therefore 

 the solution expresses determinately that there is no motion ; 

 that is to say, there cannot be any "free oscillation " of the as- 

 sumed type and period, § 3, (3)*, except in the critical case 

 alluded to. This critical case is the case in which the deno- 

 minator of the expressions for C and K vanishes, or 



«(/«•") "«m ( ; 



Then (54) gives infinite values to C and K unless L is zero; 

 and if L is zero, (53) gives 



C_ B0*)_ /3(^') , 



K - u(v!')- .&*')' • * • (™> 

 thus determining the ratio of C to K but leaving the magni- 

 tude of either indeterminate. 



14. The problem of finding all the fundamental modes of 

 free oscillation of our supposed zonal sea is solved by giving to s 

 the values 0, 1,2, &c, and for each value of s treating (57) as a 

 transcendental equation for the determination of a. After the 

 manner of Fourier and Sturm and Liouville, it may be proved 



* This equation defines perfectly the configuration of the assumed 

 motion, and specifies also that its period is — , or its " speed " <r. 



