202-203] FREE OSCILLATIONS. 337 



In the case of symmetry about the axis ($ = 0), we have, in 

 real form, 



oa(<rt + e) .................. (12), 



where K is determined by 



JV(*a)=0 ........................... (13). 



The corresponding values of <r are then given by (6). The 

 free surface has, in the various modes, the same forms as in 

 Art. 187, but the frequencies are now greater, viz. we have 



(14), 



where <T O is the corresponding value of a- when there is no 

 rotation. It is easily seen, however, on reference to (3), that the 

 relative motions of the fluid particles are no longer purely radial ; 

 the particles describe, in fact, ellipses whose major axes are in the 

 direction of the radius vector. 



When s > we have 



f=4/i(*,r).cos(<rf + *0+e) ............... (15), 



where the admissible values of K, and thence of <r, are determined 

 by (9), which gives 



35/.(*,a) + ^/.(*,a)=0 ............ (16). 



The formula (15) represents a wave rotating relatively to the 

 water with an angular velocity <r/s, the rotation of the wave being 

 in the same direction with that of the water, or the opposite, 

 according as <r/n is negative or positive. 



Some indications as to the values of o- may be gathered from a graphical 

 construction. If we put Ka 2 =x, we have, from (6), 



er/2=(l+a/0)* .............................. (i), 



where j3=4n 2 a%A .................................... (ii). 



It is easily seen that the quotient of 



sf 8 ( K ,a) by a^f 8 ( K) a) 



is a function of a 2 , or #, only. Denoting this function by $ (a?), the equation 

 (16) may be written 



The curve y=-^(x) .................................... (iv) 



L. 22 



