300 Vibrations of a Rectangular Sheet of Rotating Liquid. 



As was to be expected, the small terms in (14), (15) are in 

 quadrature with the principal term. The success of the 

 approximation requires that the frequency of revolution be 

 small in comparison with that of vibration. 



If y 1 be such that cos {\iry\lx\) vanishes, or even becomes 

 very small, the solution expressed in (14), (15) fails. This 

 happens, for example, when the boundary is square, so that 

 y Y -=.oc x . The inference is that the assumed solution (3) does 

 not, or rather does not continue to, represent the facts of the 

 case as a first approximation. 



From the principles explained in the previous paper, or 

 independently, it is evident that in the case of the square (3) 

 must be replaced by 



w = cos^, v= +icosy, . (17) 



corresponding to which 



?=^(-isin# + sin#) (18) 



These values satisfy all the conditions when there is no rota- 

 tion, and <t = s/ (gh), as in (4). For the second approximation 

 we retain these terms, adding to them u\ v\ £', which are to 

 be treated as small. So far, the procedure is the same as in 

 the formation of (6) , (7) ; but now we must be prepared for 

 an alteration of a from its initial value <t by a quantity of 

 the first order. Hence, with neglect of n 2 , 



- . dV 



i(<r— cr ) cosa 1 -}- iatfx' -\-%ii cos 3/= — g-^ . . (19) 



(XX 



— . dV 



+ (a— cr ) cos y + ia v' + 2n cos x — —q-r- • • • (20) 



ay 



These equations are the same as would apply in the absence 

 of rotation if we suppose impressed forces to act parallel to u 

 and v proportional to 



i(o- — a ) cos cX+2ni cosy, .... (21) 



=F(o-—c- ) cosy + 2n cos #, . . . . (22) 

 respectively. 



The complete solution of (19), (20) to the first order of n 

 would lead to rather long expressions. The point of greatest 

 interest is the alteration of frequency, and this can perhaps 

 be most easily treated by a simple mechanical consideration. 

 The forces given in (21), (22) must be such as not wholly to 

 disturb the initial motion (17) with which they synchronize. 

 Accordingly (21) must be free from a component capable of 

 stimulating a vibration similar to u = cos a\ and in like 



