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= -coy, v = co.r, w=Q, where CD is a function of r and t only, we 



find 



d<*_ 



dt. i- u vua i /-\ 



o?o> 



Eliminating jt?, we obtain 



2*? + ,^ 

 dt drat 



The solution of this is 



where F and / denote arbitrary functions. Since o> = when 2 = 0, we have 



and therefore 



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\fdt=^ or 

 \=p.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 fjt,=yC/2Trp. 

 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 (12 = - /t tan" 1 yjx), though a ' cyclic ' one. As a rule, in electro-magnetic 

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



