and Magnetic Field Constants. 



205 



distance from C of the near end of the cylinder, and denote 

 the length of the cylinder by Z, we get 



V = 2ir P a f e-^(l-e-*)J 1 {\a)J [\y) C ~ 



jo 



X- 



Z = 2 7 rpa^e-^Xl-e-^J 1 [\a)J {\y)~, ±. (14) 



dX 



11 = 2Trpa I e- Xz (l-e~ xl ) J x (Xa) J^Xy) r -'\ 

 Jo x 



€ase (4). A right cylindrical array of equal doublet-disks 

 exists closely and uniformly packed from to 0' 

 (fig. 2) with their common axis OC produced back- 

 ward : to find O, Z, and R for this arrangement. 



Let again the length of the cylinder be /. The arrange- 

 ment is obviously equivalent to a positive disk of surface 

 density cr at 0,- and a negative disk of numerically the 

 same density at 0'. Hence, if z be the distance of P from 



we get ut once 



iX 

 X' 



,1 e-^(l-e~ xl )J 1 (Xa)J (Xy)dX, > 



ziroa 



(1:5) 



n = 2ir<ja\ e- x "{l-e- xl )J l (Xa)J l (Xi/)dX. \ 



3. In the four cases above specified we can find the total 

 flux of force across a circle of radius b with its centre at <J. 

 Multiplying in (7) under the sign of integration by ydydO 

 and integrating from y — to y = b, and from = to Q = 2tt, 

 we find : 



dX 

 X 



For case (1), 



f f \ydydd = 4ttWj( e~ Kx J^Xa) J,(Xb) ^ 



For case (2) we find in the same way, 



) ZydydO = ±Trhnab\ e- Xz J\(\a) J^XfydX 

 *■ o ^o Jo 



(16) 



(1TJ 



