CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNESS. 321 



And therefore we obtain a solution for steady motion in the form 



L ffi> 4. <!B\ - ?- !L 



^te^dv ' ' dv) " 2-rrdz dv ' 

 This is the problem of ARAOO'S disc. The result agrees with MAXWELL, 668 (24). 



Concerning Similar Shells Enclosing One Another. 



38. Any such shell may be conceived as made up of a series of infinitely thin shells 

 successively enclosing one another. We will consider the case of similar shells, which 

 shall be defined as follows : 



Let S be a homogeneous function of x, y, and z of positive degree, the nth. Then 

 we may form about the origin a series of surfaces whose equations are S = c*, where c 

 denotes the linear dimensions. 



We may call the shells concentric. 



Points at which the same radius vector from the origin cuts the surfaces are 

 corresponding points. 



The space between two neighbouring surfaces of the series is a shell. Two shells 

 are similar when dc varies as c ; that is, the thickness of the shell at corresponding 

 points varies as c. 



If in two similar shells of a series all similarly situated the current at every point 

 in one is parallel to, and bears a given ratio to, the current at the corresponding point 

 in the other, the systems of currents are similar or corresponding. 



The ratio last mentioned may be any power of c. If it be given, then <f> in the one 

 shell is equal to < at the corresponding point in the other multiplied by a power of c. 



39. We can now compare the value of certain functions in corresponding systems of 

 currents. 



Firstly, 



Secondly, 



_ 

 Q = -- zr* - in all cases varies as the linear dimensions. 





K = QT varies inversely as the square of the linear dimensions. 



These results are independent of the form of S or <, or the ratio between the 

 currents at corresponding points. 



Thirdly, if we make 



u, v, w vary as c", 

 then 



U, V, W will vary as c" +1 . 



n and <f> will vary as c* + *. 



MDOCCLXXXVIII. A. 2 T 



