328 TIDAL WAVES. [CHAP. VIII 



assume unequal values for the two oppositely signed roots ( X) of 

 any pair. If we put X = i(r, the general symbolical value of q s 

 corresponding to any such pair of roots may be written 



q. = OA r , (ia) & 

 If we put 



we get a solution of our equations in real form, involving two 

 arbitrary constants K, e ; thus 



qi = F l (<7 2 ) . K cos (at + e) - cr/x (a 2 ) . K sin (<rt + e), \ 

 q, = F z (o- 2 ) . K cos (at + e) - af, (<7 2 ) . K sin (at + e), I 

 q 3 = F s (a*).Kcos(at + e)-af 3 (a*).Ksin(at + e), j" 



The formulae (14) express what may be called a 'natural 

 mode ' of oscillation of the system. The number of such possible 

 modes is of course equal to the number of pairs of roots of (10), 

 i.e. to the number of degrees of freedom of the system. 



If f, 77, f denote the component displacements of any particle 

 from its equilibrium position, we have 



_ dx dx 



~ j77 *?i ~J~ ~TT */2 ~H ) 



.(15). 



dz dz 



% = j Qi + -J- ^2 + - 

 d<?l ^?2 



Substituting from (15), we obtain a result of the form 



.(16), 



where P, P', Q, Q', R, R are determinate functions of the mean 

 position of the particle, involving also the value of a, and there- 

 fore different for the different normal modes, but independent of 

 the arbitrary constants K, e. These formulae represent an elliptic- 



* We might have obtained the same result by assuming, in (5), 



qi = Ai&+ t \ to = At#<*+<\ <Z 3 = ^ i(<r * +e) , ...... . 



where A lt A^, A a , ... are real, and rejecting, in the end, the imaginary parts. 



