89.] MOTION WITHIN A ROTATING EQUILATERAL PRISM. 93 



Let us choose the constants so that the straight line x = a may be 

 part of the boundary. The conditions for this are 



Aa 3 + \&amp;lt;*tf =C, - SaA + io&amp;gt; = 0. 

 Substituting the values of A, C hence derived in (47), we have 



x*-a 5 - 3^ 2 + 3a (x* - a? -f y 2 ) = 0. 

 Dividing out by x a, we get 



a 2 + 4ax + 4a 2 = 3/, 

 or x + 2a = + ^3 . y. 



The rest of the boundary consists then of two straight lines 

 passing through the point ( 2a, 0), and inclined at angles of 30 

 to the axis of x. The complete boundary, therefore, is composed 

 of three straight lines forming an equilateral triangle, the origin 

 being at the centre of gravity. 



We have thus obtained the formulas for the motion of the 

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

 when the latter is set in motion with angular velocity co about an 

 axis parallel to its length and passing through the centre of 

 gravity of its section ; viz. we have 



^ = - r 3 cos 30, &amp;lt;/&amp;gt; = - - r 3 sin 30, 



CL CL 



where 2&amp;gt;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 cases (a), (6) are mere 

 adaptations of two of M. de Saint- Venant s solutions of the latter 

 problem. 



(c) With the same notation as in Art. 88 (d), let us assume 

 We have, from (41), 



# 2 + y * = ^ (e^ - 2 cos 

 Hence (46) becomes 



Ce~*&amp;gt; cos 2f + Jeoc 2 (e-* - 2 cos 2f + &amp;lt;r 2r &amp;gt;) = const. 



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



