ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 169 



3v 3y 3y 3y 3y 3y 



II " tl ^_ V XV 41 ft ^V ^^ ~ ^V 3|* *y __<i __ *t r~ 



(i ~ * -\ > V 2 v * T~ i '" ^ '/ -\ ! 



y cz dy dx fa ' 3y y ox ' 



where x is an arbitrary function of a;, y, z independent of t. These solutions make 

 ux + vy + wz 0, and therefore involve no deformation of the surface. If we refer 

 them to polar co-ordinates, they are equivalent to 



siu 6 

 or 



u = ^ 



1 tf ti<t> 



2~ 



fj?) V = (1 

 This solution becomes identical with that found above if 



X = - ,; 



For the special case s = 0, the values of \ corresponding with the roots of the 

 second class will be zero, and the corresponding types of motion steady, even when 

 is finite. The types of steady motion will however involve a deformation of the 

 free surface in all cases where the angular velocit} 7 of rotation is different from zero. 

 These cases have been fully discussed in Part I.. 14. 



12. Forced Oscillation*. General Analytical Solution. 



The problem of the. force'd oscillations involves the determination of the quantities 

 C^ in terms of y". from the equation (31) or (40). Dealing first with the case of 

 uniform depth, and supposing that the disturbing potential involves only a single 

 term y*,P^ (p.) e i(M+i *\ we have to solve the simultaneous equations 



= o, 



with the condition that L/ C* = 0. 



r=*> 



From these we deduce, as in 5, that 



C> 



r-2 



VOL. exci. 



