160 Sir William Thomson on the 



a and a being the radii of the hollow cylindric interior, or free 

 boundary, and of the external fixed boundary, and r a the value 

 of r when r is approximately equal to a. The condition 

 T=c/r simplifies (9) and (14) to 



1 dw , ho 



P— ~r, and t= — , . . 



m ar mr J 



. . (23) 



dhv 1 dw ftw „ , 

 dr l r dr r l 



• • (24) 



and by (7) we have 



i / . k\ 



•57= — 1 )i : It 



m \ r/ 

 Hence 



w=CI i (mr) + Cfa(mr); . . . 



. . (25) 

 . . (26) 



and the equation of condition for the fixed boundary (radial 

 velocity zero there) gives 



Gl / i(ma) + €¥ i (ma)=0 (27) 



To find the other equation of condition, we must first find an 

 expression for the disturbance from circular figure of the free 

 inner boundary. Let for a moment r, be the coordinates of 

 one and the same particle of fluid. We shall have 



= ^6dt; and r = §rdt + r 0j 



where r denotes the radius of the " mean circle" of the par- 

 ticle's path. 



Hence, to a first approximation, 



e=$; (28) 



and therefore, by (6), 



(ic\ 

 n ^ it; 



whence p 



o 



cos 7iiz cos (nt—i0). . (29) 



Hence the equation of the free boundary is 



n — ico 



where 



r==a--^f^cos(^-^), .... (30) 



o=^ (31) 



