ivith the Flow of Electricity in a Plane. 385 



§ 13. A few special cases of this may be recorded. 

 When r x = r 2 , it becomes 



1 , (r cosna— cosnfi \ ,~x 



7tkB °\p n*J (sin ncc . sin w/3 ' ' * ^ 



If at the same time a be made zero and /3 = 0, we ought to get 

 (4) ; but we must remember that a is not actually to vanish, 



it is to become very small and equal to J-, which is a sort of 



compromise between -, the greatest value it could vanish with, 



and -, the least. So putting 



= 2? '-'-& 



L_i (n.l\= —\ ^ 



(6) becomes 



2r 

 which agrees with (4). 



Going back to i\ and r 2 unequal, let us put both poles on 

 OX — that is, write 



a =0=^and/3 = 0=^ 

 2r x 2r 2 



(5) will become 



1 / r\-rl V x 7 x 



^s^Wpvr^- 1 / 5 KJ 



and this reduces to (2) when r is made infinite. 



Similarly if the poles were placed one on OX and the other 

 on OY, we should get 



* log (r}^f_ ; (8) 



and this reduces to (3) when r 1 =r Q + c = co , and 0= - = 0. 



One more case is worth mentioning, viz. when one of the 

 poles, say B, is on the angle of the wedge at 0. To give 

 this, r 2 — p and (0 being any thing between and 0, w/3 will be 



7T 



any thing between and ir, say -x) sinw/3 = l, so 



~~ 2-ttkB °& \p*n+i ' 2n sin nu)' } ' 



Phil Mag. S. 5. Vol. 1. No. 5. May 1876. 2 D 



