195 197] DIELECTEICS AND MAGNETIZABLE BODIES. 375 



This gives the case of the infinitely long circular cylinder, for 

 which, as we have previously found, the longitudinal demagnetiz- 

 ing factor vanishes, while for transverse magnetization it is equal 

 to 2?r. 



When e 1 the expressions for the oblate ellipsoid give L = 4?r, 

 M = N = 0. This gives us the case of a disk magnetized normally, 

 for which the demagnetizing factor is the largest possible, namely 

 4?r, or parallel to the faces, when the demagnetizing factor is zero. 



For a long prolate ellipsoid, for which e is nearly unity, we 

 may conveniently use an approximate formula. Putting m = a/b 

 for the ratio of the length to the diameter, since 



a m 

 we have approximately 



(6) M=N= (log 2m - 1). 



fti/ 



A table of values of the demagnetizing factor is given by 

 Ewing*, and a larger one by du Boisf. 



197. Magnetization of Hollow Cylinder. We shall now 

 consider a few cases of induction in which the induced magnetiza- 

 tion is not uniform. In the first case let us consider the uniplanar 

 problem of the transverse magnetization of an infinite homo- 

 geneous circular cylinder, placed in a field such that the lines of 

 force are the intersections of cylindrical surfaces with planes 

 perpendicular to the generators of the cylinder. If the cylinder is 

 circular the method of development in series of circular harmonics, 

 94, gives the general solution of the problem. 



Let the cylinder be hollow, the inner radius being b and the 

 outer a, the inductivity of the cylinder being /i 2 , and of the space 

 within and without /4 lt Let the undisturbed field, as before, be 

 represented by F with potential F , while the field due to the 

 induced polarization is Fi with the potential F^. We shall 

 suppose that the bodies producing the field lie outside the 



* Ewing, "Magnetic Induction in Iron and other Metals," p. 32. 



t du Bois, "Magnetische Kreise, deren Theorie und Anwendung," p. 45. 



