374 Mr. 0. J. Lodge on some Problems connected 



ment ; these are the subjects I propose to develop in the pre- 

 sent communication. 



§ 1. Imagine a plane sheet, some portions having infinite 

 conducting-power, other portions consisting of ordinary matter, 

 and again other portions possessing infinite resisting-power. 

 The whole extent of any of the infinitely conducting regions 

 will have a uniform potential ; and hence the bounding line 

 between any such region and the ordinary matter of the sheet 

 must be an equipotential line. Further, if one defines a line 

 of flow or stream-line as a line across which no electricity passes, 

 the boundary line between the ordinary matter of the sheet and 

 an infinitely resisting region must be a line of flow, for no elec- 

 tricity can possibly get across it. Such a sheet will therefore 

 contain a certain number of flow-lines and a certain number of 

 equipotential lines. Moreover let the potential of any one of the 

 conducting regions be maintained higher than that of some 

 other one, then in general a uniform flow of electricity will 

 take place throughout the whole of the ordinary matter of the 

 sheet, which will therefore contain additional stream-lines and 

 equipotential lines, whose arrangement will depend on the situa- 

 tions and potentials of the various non-material regions. 



Some general properties of the equipotential and stream- 

 lines were pointed out in § 35 of the paper referred to. One 

 of these properties is, of course, that the lines of one system cut 

 the lines of the other orthogonally ; a more general statement 

 is the following. If a flow-line passes n times through a given 

 point of the sheet, it will form at that point an equiangular 



IT. 



pencil whose 2n rays meet each other at an angle — ; and if no 



pole exists at the meeting-point it will be a point of no flow, 

 and an equipotential line will also pass n times through the same 

 point, making an equiangular pencil the same in all respects as 



the former, but turned through an angle of ~— , so that its rays 



bisect the angles of the first pencil. The ordinary orthogonal 

 section is the special case of this when 71 = 1. The special case 

 ri=oo is obtained if, in the sheet imagined above, one of the 

 conducting regions touches one of the resisting regions along 

 a line ; when this happens, the line separating the two may be 

 regarded indifferently as a stream-line or as an equipotential 

 line. 



There are then two ways of making any given line a stream- 

 line — viz. either by arranging the non-conducting regions so 

 that the line shall bound one of them, or else by arranging the 

 conducting regions and their potentials in such a way that 

 none of the flow of electricity shall take place across any por- 



