us 



Mr. S. H. Burbmy. On the Induction of [May 3 y 



2T 



where is the current function, Q the magnetic potential, and dQ/dv 

 the rate of its variation per unit of length of the normal. 



7. The magnetic induction due to the sheet with current function 

 is the same as that due to a magnetic shell of strength over the 

 surface at all points not within the substance of the shell. 



8. Given any magnetic field external to a surface, S, there exists a 

 determinate system of magnetic shells over S having at all points 

 within the surface magnetic potential equal to that of the external 

 field. 



9 and 10. Therefore also a system of currents over the surface 

 having the corresponding property, called the magnetic screen. 

 Example of a sphere. 



11, 12, and 13. If the function yjr satisfy the conditions 



dfjdv — ZF + mG + riR on S, 

 = within S, 



then F = dyjr/dx, &c, if F, G, H be the components of vector 

 potential due to the external system and its magnetic screen, yjr is 

 called the companion function to F, G, H. 



14 — 17. Solution of the problem of induction in the absence of 

 resistance by Lagrange's equations, where the external system varies 

 continuously, in the form — 



= 



dt d$ 



2 T = j| ft (a + g)* 5 . 



where O , Q , and S relate to the external system, and 0, Q, and S to 

 the induced currents on S. 



18. This gives at all points within S 



d(F Q + F) _ d(jr + jr) . 

 dt ~~ dx ' *' 



where ^ s * ne companion function to < ^ ) , ~^jTi ai1 ^ an( ^ 



dF dG , dB. 



