32 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II 



30. As a final example, we will take one suggested by the 

 theory of electro-magnetic rotations. 



If an electric current be made to pass radially from an axial wire, through 

 a conducting liquid (e.g. a solution of CuS0 4 ), to the walls of a metallic 

 containing cylinder, in a uniform magnetic field, the external forces will be 

 of the type 



Assuming u= -to?/, v = o&amp;gt;.r, w = 0, where o&amp;gt; is a function of r and t only, we 

 find 



d&amp;lt;0 2 _ \lX 



Eliminating p, we obtain 



2 dt +r drdt = 

 The solution of this is 



where F and / denote arbitrary functions. Since o) = when t = Q, we have 

 and therefore 



^oo-^(o) x 



where X is a function of t which vanishes for = 0. Substituting in (i), and 

 integrating, we find 



Since p is essentially a single-valued function, we must have d\/dt=n, or 

 \ = fj.t. Hence the fluid rotates with an angular velocity which varies 

 inversely as the square of the distance from the axis, and increases con 

 stantly with the time. 



* If C denote the total flux of electricity outwards, per unit length of the axis, 

 and 7 the component of the magnetic force parallel to the axis, we have /*= 7(7/2717). 

 For the history of such experiments see Wiedemann, Lehre v. d. Elektricitat , t. iii. 

 p. 163. The above case is specially simple, in that the forces X, Y, Z, have a 

 potential (ft = - /j. tan&quot; 1 y/x), though a cyclic one. As a rule, in electro -magnetic 

 rotations, the mechanical forces X, Y, Z have not a potential at all. 



