1888.] Electric Currents in Shells of small Thickness. 149 



19. If therefore — dF/'dt, &c, are to be regarded as components of an 

 electromotive force, notwithstanding their derivation from a potential 

 within S, they will produce on S a distribution of free electricity 

 having potential — (^ + ty), and forming a complete electric screen. 



20. There is no energy of mutual action between the electrostatic 

 system, if it exists, and the electric currents, because 



IK***-*)—' » 



21 and 22. The effect of resistance generally. 



23. Definition of self-inductive current shells, viz., those for which 

 the values at any time, t, of the component currents, u, v, vj, &c, are 

 found from their values at a given epoch by multiplying by e~ kt where 

 X is constant. 



24. Investigation of the condition which 0, the current function ., 

 must satisfy in order that a current shell may be capable of being 

 made self-inductive. 



25. If this condition be satisfied, the thickness of the shell which 

 makes it self-inductive is determinate, the material being supposed 

 uniform. 



26. And X. varies inversely as the thickness. 



27. General property of self -inductive shells in presence of a cor- 

 responding magnetic field whose potential is fi expressed by the 

 equation — 



t+f +">-»■ 



at all points within the shell, or on the opposite side of it to the 

 inducing system. 



28. Example (1), when dQjdt = constant. 



29. Example (2), when Q = A cos Xt and A- constant. 



30. Some consequences deduced from the last example. 

 Examples of self -inductive shells, viz. : — 



31. Spherical shell. 



32. Solid of revolution about the axis of z } being a function of z 

 only. 



33. Any surface if be a function of z only and a function of X 

 and y only. 



34. Example, an ellipsoidal shell. 



35. Case of an infinite plane sheet as made self-inductive in certain 

 cases. 



36. Case of an infinite plane sheet when not self-inductive. Arago's 

 disk. 



37 — 40. Self-inductive shells bounded by a surface, S, when S is a 

 homogeneous function of x, y, and z. 



m 2 



