﻿458 Mr. W. M. Hicks on the Self-Induction 



and bounded by this circle. Let w, w ! denote the velocities 

 at P parallel to NP and Oz. Then 



■jL. dp = tlux up through an annulus whose radii are 

 d P PNandPN + rf/o, 



= 2irpdp u/, 



d"\lf 



~r-dz = flux through cylindrical tyre (breadth dz) 



towards N, 



= —27rpdzw; 

 whence 



2irp dz ' "~ 27T/3 dp 



2. Consider a current flowing through an elementary ring 

 whose cross section is dp dz and in which the current- density 

 is cr. Then the circulation taken round this 



= 47r x current = 47r<r dp dz. 



But the circulation 



(dw dw 1 



whence 



or 



= {£-^) d P dz=4 ^ <rd f >dz; 



2tt P dz* + dp\2irp dpi ' 



d*yfr d*+ ld+ . 



At points where there is no current the right-hand side is of 

 course zero. 



3. To find the force-flux we shall require two functions, 

 one (yjr') for space outside the ring, and the other (yfr) for 

 space inside. The conditions which they have to fulfil are as 

 follows : — ■ 



yfr' must be finite at all points outside the ring, be zero at 

 the point 0, and satisfy the equation 



dz 2 + dp* p dp ~ U - 

 Using toroidal functions, these conditions are satisfied by 



yfr' = —jr^ r 2R» (A n cos nv + Aj» sin nv) . 



V [Kj — C) 



