270 TIDAL WAVES. [CHAP. VIII 



important illustrations will present themselves in the theory of the 

 tides*. 



When a is very great in comparison with a s , the formula (10) 

 becomes 



C 

 q s = -- ~cos(&amp;lt;rt + 6) ............... (13); 



the displacement is now always in the opposite phase to the force, 

 and depends only on the inertia of the system. 



If the period of the impressed force be nearly equal to that of 

 the normal mode of order s, the amplitude of the forced oscillation, 

 as given by (12), is very great compared with q g . In the case of 

 exact equality, the solution (10) fails, and must be replaced by 



q s = Bt sin (at + e) ..................... (14), 



where, as is verified immediately on substitution, B = C s /Zaa s . 

 This gives an oscillation of continually increasing amplitude, and 

 can therefore only be accepted as a representation of the initial 

 stages of the disturbance. 



Another very important property of the normal modes may be noticed, 

 although the use which we shall have occasion to make of it will be slight. 

 If by the introduction of constraints the system be compelled to oscillate 

 in any other manner, then if the character of this motion be known, the 

 configuration at any instant can be specified by one variable, which we will 

 denote by 6. In terms of this we shall have 



fc-*t4 



where the quantities B 8 are certain constants. This makes 



.)0 .............................. (i), 



)P ............................ (ii). 



Hence if 6 a cos (o-tf + e), the constancy of the energy (T+ F) requires 



Hence o- 2 is intermediate in value between the greatest and least of the 

 quantities c B la a ; in other words, the frequency of the constrained oscillation 

 is intermediate between the greatest and least frequencies corresponding to 

 the normal modes of the system. In particular, when a system is modified 

 by the introduction of any constraint, the frequency of the slowest natural 

 oscillation is increased. 



* Cf . T. Young, &quot; A Theory of Tides,&quot; Nicholson s Journal, t. xxxv. (1813) ; 

 Miscellaneous Works, London, 1854, t. ii., p. 262. 



