312 MB. S. H. BURBURY ON THE INDUCTION OF ELECTRIC 



Now if the current function <f> be given for a surface S, U, V, W, F, G, H, and x 

 are determinate. 



Also the resultant of F dyjdx, G d-)(Jdy, H dyjdz is necessarily in the 

 tangent plane. But it is not necessarily in the same direction in that plane with the 

 resultant current. 



But the equations (B), or 



F - dxldx G - d x ldy _ H - 

 U V~ W 



cannot be satisfied unless the resultant of F dyjdx, &c., is in the same direction in 

 the tangent plane with the resultant current. They, therefore, express a necessary 

 condition which the current function, <, must satisfy, in order that the system may be 

 capable of being made self-inductive. Evidently they express only one condition at 

 every point, that a line known to be in the tangent plane shall have a particular 

 direction in that plane. It may be put in the form of a partial differential equation 

 to be satisfied at every point on S, namely, 



.. 



dxj dx \ dy j dy \ dz / dz 



Since, if tj> be given, ^ is determinate, this is a partial differential equation in <j> only. 

 We may assume that, inasmuch as there are as many disposable quantities, namely, 

 the values of <f> at every point, as there are conditions to be fulfilled, there must for 

 every surface S be one or more solutions. As we shall see, if S be a sphere, and in 

 certain other special cases, there are many. 



26. If this necessary condition for <f> be fulfilled, it makes the three quantities 



F dx/dx G dx/dy H 



U V W 



equal to each other at every point. But they will generally differ in values from 

 point to point on the surface. 



Let each of them be denoted by Q. Then Q is a function depending on the form 

 of S and <f> and the position on the surface. 



Then in order that the system, with <f> so chosen, may actually be self-inductive, we 

 must so regulate the thickness, h, of the shell, as that 



at every point, K being a constant. 



