of a Current running in a Cylindrical Coil. 213 



to any plate, AB, of radius a is of the form 



a.<\>(k,6), (24) 



where </>(&, 6) is the coefficient of a in (15), and <f>(k, 6) is a 

 function of 6 and 6', the angles PAB and PBA. Hence if 

 we take a plate of radius OQ, and from B draw B^ parallel to 

 QP and meeting OP in q, the potential of this plate at P is 

 to the potential of the plate AB at q as OQ is to OA ; for, if 

 AR = BQ, the angles <?BA and </AB are equal, respectively, 

 to PQR and PRQ. Hence, if r=OQ and V is the potential 

 at q due to the plate AB (of radius a), the resultant potential 

 at P due to the series of plates of radii extending from OB to 

 OT is 



s*- v * (25) 



the points q on OP ranging from t to P, where B£ is parallel 

 toPT. 



Of course any plate of the series may be taken instead of 

 AB as the reference plate. 



Thus, the resultant potential, due to all the plates, is calcu- 

 lated from values of the potential of any one plate at a series of 

 points ranged along the radius vector OP. 



Pass now to the consideration of the practical problem in 

 hand, viz., the potential at P due to a coil of depth 00', i. e., 

 we have to consider the whole of the spaces BTT'B' and 

 ASS' A' filled with wire traversed by a current of strength i. 

 We have already seen that we have to subtract from the 

 potential at P due to a series of uniform attracting plates, 

 each of surface -density m, ranging from the radius OB to the 

 radius OT, the potential at P due to the lower series, each of 

 surface-density m, and ranging from radius OB' to radius OT'. 

 It merely remains to express m in terms of current-density. 

 If C is the total quantity of current traversing (at right 

 angles to the plane of the paper) a unit area (square centi- 

 metre) of the space BTT'B', the quantity flowing in a filament 

 of depth dy and breadth clr is Gdydr. Now this filament is 

 replaced by the magnetic shell of radius r and thickness dy ; 

 and since we know that the strength of the shell is equal to 

 the current in the filament, we have m . dy = Cdydr, .'.m = Cdr; 

 hence (25) becomes, from (15), 



±C§r.cf>(k,d)dr, (26) 



which is the potential due to the upper series of plates, 

 OB, . . . OT. 



This quantity may be graphically represented and calcu- 

 lated as follows. Let a very close series of curves representing 



Phil. Mag. S. 5. Vol. 37. No. 225. Feb. 1894. Q. 



