277-278] RETARDATION OF SPRING TIDES. 503 



m, . , , AfT + A v , A(T A v . ...... 



This ranges between . , and - -- r ,- ........................ ( vin)*. 



A-f A A A 



The above is the well-known explanation of the phenomena of the spring- 

 and neap-tides f ; but we are now concerned further with the question of phase. 

 In the absence of friction, the maxima of the amplitude C would occur whenever 



(&amp;lt;r r -&amp;lt;r)t = 2mir + e - e , 



where in is integral. Owing to the friction, e and e are replaced by quantities 

 of the form e - f x and e - e/, and the corresponding times of maximum are 

 given by 



(a- &amp;lt;r)t (e/ - ej) = 2wi7r + e - e , 



i. e. they occur later by an interval (e/ - e 1 )/(o- - o-). If the difference between 

 a- and a- were infinitesimal, this would be equal to dejdo: 



In the case of the semi-diurnal tides in an equatorial canal we have &amp;lt;r = 2ft , 

 f 1 = 2x, whence 



_ 

 do- dri 



278. Returning to the general theory, let q^ q. 2 ,... be the 

 coordinates of a dynamical system, which we will suppose subject 

 to conservative forces depending on its configuration, to motional 

 forces varying as the velocities, and to given extraneous forces. 

 The equations of small motion of such a system, on the most 

 general assumptions we can make, will be of the type 



where the kinetic and potential energies T, V are given by ex 

 pressions of the forms 



2 g^ 2 + ............ (2), 



2 g 1 ^ 2 + ............ (3). 



It is to be remembered that 



d rs = d sr , C r8 = C sr ........................ \*)&amp;gt; 



but we do not assume the equality of B rs and B sr . 



If we now write 



..................... (5), 



* Helmholtz, Lehre von den, Tommpjindunyen (2 Aufl.), Braunschweig, 1870, 

 p. 622. 



t Cf. Thomson and Tait, Natural Philosophy, Art. 60. 

 $ Cf. Airy, &quot;Tides and Waves,&quot; Arts. 328... 



