456 Messrs. G. C. Foster and 0. J. Lodge on the Flow 

 which gives 



d^Va' + p 2 ^ — — £g , d 2 =Va* + pl= ^ , 



and a the same as before. Hence (7) may be written 



R ''=2^ 



2p,p 2 



25. If instead of considering the whole extent of the sheet we 

 confine our attention to that part of it which is contained be- 

 tween any two lines of flow (see Plate IX.), its resistance will be 

 given by any of the above formula if instead of 2tt in the deno- 

 minator we put the angle, say 7, which the lines of flow forming 

 the boundaries make with each other at either pole; for the 

 spaces between every pair of consecutive flow-lines convey equal 

 currents; and since the difference of potential between their 

 ends is the same for all, they all have the same resistance. Con- 

 sequently the resistance of a part of the sheet made up of such 

 spaces is inversely proportional to the number of them which 

 compose it. Now, using as previously a to stand for the angle 

 which two consecutive lines of flow make with each other at the 

 poles, the total number of spaces into which the whole sheet is 



2tt 

 divided is — , and the number of spaces lying between lines of 



flow whose angle at the poles is 7, is - ; hence, if R is the resist- 



ance of the part of the sheet bounded by these lines, and R the 

 resistance of the whole of the sheet, we have 



R 7' 



In the case of two flow-lines which form arcs of the same 

 circle drawn through the two poles, we have 7 = 77-, and therefore 



R y =2R. 



Hence, if a small equipotential circle is drawn round each pole, 

 the resistance of the part of the sheet lying between these circles 

 and contained within any circle drawn through the poles is the 

 same, being equal to twice the resistance of the unlimited sheet 

 between the same two equipotential circles, and also equal to the 

 resistance of as much of the unlimited sheet as would be left 

 after cutting out the portion bounded by any circle through the 

 poles. Consequently if p is the common radius of the small 



(12) 



