ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 185 



constituents, approximates to one day, and attains this as a limiting value when 

 a = 0. Hence, as diminishes, or h increases, the largest root of the second class 

 must pass through the value '92700, thus rendering one of the lunar diurnal 

 constituents infinite. This accounts for the rapid increase in the coefficients in the 

 series given above for these tides as h increases. 



The roots of the first class must however all be greater than unity, no matter 

 how great the depth may be. Since they all decrease with the depth, they must 

 approach finite limiting values greater than unity as the depth diminishes to zero. 



The tides of rigorously diurnal period will become infinite when the depth is 

 variable if / assumes the value 



j tO'll' 



so that with this value of I we may anticipate that there will be a period of 

 free oscillation exactly a sidereal day in duration. Now the above value of / will 

 require that the surface of the solid earth should be rigorously spherical in order 

 that the free surface of the ocean may be an equipotential surface under gravity and 

 centrifugal force. It is easy to see why in this case a free oscillation of rigorously 

 diurnal period must exist. For if the water be set in rotation as a solid body about 

 an axis not rigorously coincident with the rotation-axis of the solid earth, and the 

 form of the free surface be adjusted for equilibrium under centrifugal force about the 

 new axis of rotation, there will be no forces acting which tend to modify this state 

 of motion, and it will continue permanently, provided the system be free from 

 friction. The motion of the water will be steady in space, but it will be oscillatory 

 with a period of one day relatively to the solid earth. 



It is easy to verify that in the forced oscillations of rigorously diurnal period the 

 motion of the water is of like character, involving no relative motion of the parts 

 and being steady in space. In this case the axis about which the rotation of the 

 water takes place lies in the plane containing the earth's polar axis and the dis- 

 turbing body. 



VOL. CXCt. A. 2 B 



