ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 

 in virtue of the differential equation (1) for P B ; and therefore by means of (5) 



(i - ^.ffi = , p. _ OL^L i) ( f o * * 



'dp 



143 



2n 



which with the aid of (6) may be expressed in the form 



- 



Let us write for brevity 



d 



AEE~(D)- 



(8)- 



Then the equation (3) may be written 





, = - n(n+ 1)P: 

 while, if a- denote any constant quantity, we obtain from (6), (7) 



(D 



(9), 



The relation between the operators D, A may be written in the forms 



(D - o/t) (D + cr/t) - (^ ~ ^-) = (1 - M 2 ) (A + r) 1 

 (D + 07*) (D - cr M ) - (# - aV 2 ) = (1 " M 2 ) (A - a) J 



which will be useful hereafter. 



2. Transformation of the Dynamical Equations for the Tides. 



The formation of the differential equations for the tidal oscillations of the ocean 

 has been fully dealt with in Part I. It is there shown ( 4) that, if U, V denote the 

 northward and eastward velocity-components in latitude sin" 1 ^ and longitude </>, 

 when the system is executing a simple harmonic vibration in period 2ir/\, these 

 velocity-components will be expressible in terms of a single function i/ by means of 

 the equations 



" 





a (X - 4 V 2 ) 



" h 



v/ ( 1 - 



2 - - 4 V) 



_ 



(X - 4w V) 



, 1 2 v 



a v/ (1 - p 2 ) (X 3 - 4a>V) 





