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 co about an axis 

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

 viz. we have 



............ (7), 



where 2 ^Sa 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- Venant'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 20 fr\ 



(a) 



C S 



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

 ^r = ^a>r 2 for = a, and the constants A 2n+l can be determined 

 by Fourier's method so as to make ty = ^coa 2 for r a. We 

 find 



)7r _ 4a - (2n+ i )7r + (2 n + 1) TT + 4a{ 



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



The conjugate expression for < is 



oi n O# /*.\ (2n+l)ir/2o 7T/9 



* \<*r<> 5J|? - SA^ g) sin (2 + 1) g. . .(10)f, 



where -4 2 ?i+i has the value (9). 

 The kinetic energy is given by 



tt *rdr ............ (11), 



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



t This problem was first solved by Stokes, "On the Critical Values of the 

 Sums of Periodic Series," 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 



