Certain Problems of Two-Dimensional Physics. 769 



Electrical Problems. 



7. The potential of a freely electrified cylinder is 

 evidently 



^ =V+ M FW_F(T) }' 



where F is so chosen that, at infinity 



$=— 2Elogr, 



E being the charge per unit length. 



Take, for instance, the case of the cylinder 



X = acos n ?7, Y = 6sin n 77, .... (4) 



in which n must be a positive integer. 



We have x + Ly = a cog » ^ _ ^ + lh sin » ^ _ ^ . 



and at P = + co , 



so that log r = ?if , 



approximately. 



Thus the potential of the cylinder bounded by the curve 

 (4) is 



0=V-2nE£ 



provided that the singular points of the transformation do 

 not fall within the field of variation of <£. If n > 1 there 



77" )77" 



are singular points at f =0,^ = 0, -^ , ?r, r-; t. e. the electric 



density is infinite at these points. 



8. The potential of a cylinder magnetized transversely 

 may be determined in the same way. 



For simplicity we shall assume that the components of 

 the intensity of magnetization are derivatives of a single 

 function J, so that 



a— I*. b— £. 



d^ Of/ 



We then have 



V 2 O = 0, V 2 (O t + 47rJ)=0, 



and on the boundary 



where Il is the external, H t the internal magnetic potential. 



