72] ROTATION OF A CYLINDER. 97 



We have thus obtained the formulae for the motion of the 

 fluid contained within a vessel in the form of an equilateral prism, 

 when the latter is rotating with angular velocity w about an axis 

 parallel to its length arid passing through the centre of its section ; 

 viz. we have 



^ = - J - ? cos 30, (/&amp;gt; = -r 3 sin3&amp;lt;9 ............ (7), 



Ct (t 



where 2 /v/3a is the length of a side of the prism. 



The problem of fluid motion in a rotating cylindrical case is to a certain 

 extent mathematically identical with that of the torsion of a uniform rod or 

 bar*. The above examples are mere adaptations of two of de Saint- Tenant s 

 solutions of the latter problem. 



3. In the case of a liquid contained in a rotating cylinder 

 whose section is a circular sector of radius a and angle 2a, the 

 axis of rotation passing through the centre, we may assume 



COS 26 / ? A(2+l,7r/2a ^0 



c^ + 2 ^ +1 (a) C S (2U + l) 2S- -&amp;lt; 8 &amp;gt; 



the middle radius being taken as initial line. For this makes 

 ty = ^cor 2 for 6= a, and the constants A^n+i can be determined 

 by Fourier s method so as to make -x/r = Jwa 2 for r = a. We 

 find 



w + 1)7r _ 4a - (^Fl)^ + (IT-n^+lSI 



......... (9). 



The conjugate expression for &amp;lt; is 



sin 26 . / r \(2n+Diry2 . 7J-0 







/ r 



(a 



where A 2n+l has the value (9). 

 The kinetic energy is given by 



a (11), 



* See Thomson and Tait, Natural Philosophy, Art. 704, et seq. 



t This problem was first solved by Stokes, &quot;On the Critical Values of the 

 Sums of Periodic Series,&quot; Camb. Trans., t. viii. (1847), Math. andPhys. Papers, t. i., 

 p. 305. See also papers by Hicks and Greenhill, Mess, of Math., t. viii., pp. 42, 89, 

 and t. x., p. 83. 



L. 7 



