168 MR. S. S. HOUGH ON THE APPLICATION OP HARMONIC 



n, while Cj +J /Dj will tend towards a finite limit. In like manner when r is greater 

 than n, C*/Dj_! will tend towards zero while Cv_,/D, will tend to a finite limit. 



Let us examine the limiting forms assumed by , U, V, when the rotation is 

 annulled. On putting n (n + 1) 2ws/X = the relations (51) give 



U N _ r ( r 4. i) _ w ( w _j_ j) = ( r _ w ) ( r _|_ n _j_ !). 



Thus from (52) we obtain, if r < n, 



T t C' 4^ _Jr + jtl2_ T/ C' r + 1 (r - n) (r + n + 1) (2r + 3) 



J 3 D' r + 1 " " ^ (r + I) 2 (2r + 3) ' D' r r* (r + s + 1) 



and if?' > ??, 



T ,_eL_ 4 2 (r - a) Ct_, (r- M )(r + + l)(2r-l) 



J to 2 IV _, " ^ 7-- (2r - 1) ' I)' r (r + ]>s (r - ) 



Hence if we retain only the most significant terms and put <aD' n = A*, we find 



' +-^-- (- + !)_ p. 1 



h %.+, ( + I) 2 (2 + 1) K+1 J 



/(I _ U 2\ ]J _ __ - A .g.XW + ^) p. 



^ ^ ^ ' n(n+ l)/t " 



+ 



P; 



" ( n "*" *; AtiM+its>\~*L n + s - Ds 4ffl- (n 



2a 



p. , i'L 3 (n - + 1) 

 '"' h ( n + i)' 2 ( 2n + !) J' 



If therefore we suppose that co reduces to zero while A' remains finite, will 

 reduce to zero, but the velocity-components will tend to finite limits given by 





2a 



in virtue of (7). 



Hence the steady motions to which the oscillations of the second class reduce 

 when the rotation is annulled involve no deformation of the free surface. This of 

 course may readily be verified by a direct method. For if u, v, w denote the 

 velocity-components referred to fixed rectangular axes, it may be seen that all the 

 conditions of the problem will be satisfied by 



