328 Transverse Oscillations of a Canal of Circular Section. 



The roots of this equation should give the frequencies of 

 different modes. It has one real root, x = 1*345, and two 

 complex ones. From # = 1*345 we obtain a= si g\a 1*160. 

 Lord Rayleigh finds for the gravest mode, a-= Vgja 1*1644. 

 If we substitute # = 1*345 in equations (6) we find 



8i« -1*764, 



S 3 = -5895, 

 S 5 =- -0538. 

 Hence 



+-g)s,»»« + g) , S,*.3« + (Js,.in»-(Q"|«.2« 



= - Q 1*764 sin 0+ (-Y 1 * 187 cos 26+ C^j ' 5895 sin 36> 



-(~Y .1982cos 4(9 - (-Y-0538 sin 5(9+ (-Y-0121 cos 66. 



The accompanying figure represents the stream-lines 

 calculated by the above formula. Since they are sym- 

 metrical about OY only one quadrant is given. Hence we 

 see that with three constants we obtain a satisfactory repre- 

 sentation of the slowest mode. 



If we take more constants we obtain an equation of higher 

 degree for a; and it seems that its other real roots may give 

 the higher modes. I have, however, not been able to discover 

 any way of discussing the general equation for x, and to 

 calculate particular cases would be exceedingly laborious. 



