206 Prof. A. Gray oh Electric 



For case (3) the result is 



\ I ZijdydO = 4:7r 2 pab \ e - Kz (l-e-* 1 ) J x (\a) J^Xb) ^ 2 (18) 



Finally, in case (4), the most important of nil, 



f f ZydydQ = lir'aabf e-^(l-e- XI ) J^J^Xb)^. (19) 

 Jo Jo Jo x 



In the electromagnetic analogue, in which a current sheet 

 takes the place of the cylindrical array o£ doublet-disks, 

 we have to replace a by N7, if the sheet be produced by 

 N circular turns of fine wire, with a current 7 in each. 



4. Comparing the R of (14) with (19), we have (he 

 remarkable result that the ^/-component of force at P 

 (a point on the circumference of the circle of radius 5), 

 due to the uniform cylindrical volume distribution of 

 density p, is to a constant identical in value with the total 

 magnetic induction through that circle due to the current 

 sheet. To convert the former into the latter value it is only 

 necessary to multiply by lirbajp. This is a consequence of 

 the reciprocal theorem that the radial force at a point P 

 (on the circumference, say, of a circle of radius U) due to a 

 uniform coaxial disk of matter of any radius a, say, is, to 

 a constant, equal in value to the magnetic induction through 

 the circle of the disk produced by a current flowing in the 

 circle of radius b. The reader may think of two circular 

 coaxial cylinders, one of radius a and length / lying to the 

 left of 0, the other of radius b and situated to the right of C 

 Either may be taken as the volume distribution and the other 

 as the current sheet or array of magnetic shells. To fix the 

 ideas, take that of radius a as the volume distribution, and 

 the other as the current sheet. Then the magnetic induction 

 through the circle of radius a, centre (fig. 2), due to the 

 current sheet, is, to a constant, equal to the value of R, 

 produced at any point of the circumference of the circle of 

 radius b, and centre C. 



Or, since the induction through the circle of radius a f 

 due to the current sheet just specified, is equal to the 

 induction through the circle of radius b y due to a current 

 sheet coinciding with the cylinder of radius a, we may 

 suppose the volume distribution and the current sheet to 

 occupy coincident cylinders. Then the magnetic induction 

 through any coaxial circle due to the current sheet is, to the 



