322 TIDAL WAVES. [CHAP. VIII 



If the sheet of water considered have as boundaries two 

 meridians (with or without parallels of latitude), say &> = and 

 o> = a, the condition that v at these restricts us to the factor 

 cos 50), and gives so. = rmr, where m is integral. This determines 

 the admissible values of s, which are not in general integral*. 



Tidal Oscillations of a Rotating Sheet of Water. 



194. The theory of the tides on an open sheet of water is 

 seriously complicated by the fact of the earth's rotation. If, 

 indeed, we could assume that the periods of the free oscillations, 

 and of the disturbing forces, were small compared with a day, the 

 preceding investigations would apply as a first approximation, 

 but these conditions are far from being fulfilled in the actual 

 circumstances of the Earth. 



The difficulties which arise when we attempt to take the 

 rotation into account have their origin in this, that a particle 

 having a motion in latitude tends to keep its angular momentum 

 about the earth's axis unchanged, and so to alter its motion in 

 longitude. This point is of course familiar in connection with 

 Hadley's theory of the trade- winds (. Its bearing on tidal theory 

 seems to have been first recognised by MaclaurinJ. 



195. Owing to the enormous inertia of the solid body of the 

 earth compared with that of the ocean, the effect of tidal reactions 

 in producing periodic changes of the angular velocity is quite 

 insensible. This angular velocity will therefore for the present be 

 treated as constant . 



The theory of the small oscillations of a dynamical system 

 about a state of equilibrium relative to a solid body which rotates 

 with constant angular velocity about a fixed axis differs in some 

 important particulars from the theory of small oscillations about 

 a state of absolute equilibrium, of which some account was given 



* The reader who wishes to carry the study of the problem further in this 

 direction is referred to Thomson and Tait, Natural Philosophy (2nd ed.), Appendix 

 B, " Spherical Harmonic Analysis." 



t " Concerning the General Cause of the Trade Winds," Phil. Trans. 1735. 



J De Causa Physicd Fluxus et Refluxus Marts, Prop. vii. : " Motus aquse turbatur 

 ex inaequali velocitate qu& corpora circa axem Terrse motu diurno deferuntur " (1740). 



The secular effect of tidal friction in this respect will be noticed later (Chap. 



XI.). 



