CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNESS. 305 



Example of Magnetic Screen. 



1 0. Let S be a sphere of radius a. Then P, the magnetic potential of the external 

 system, may, as regards its value on S, be expanded in a series of spherical surface 

 harmonics, including generally a constant term, namely, 



Hence, 



V, -2n_+J 



> i * " * 



Then <f> is the solid harmonic 



which satisfies the condition 



-~ = -p = 7 at all points on S ; 



and the value of <f> on S is 



AO 1 2n + 1 . v 



<p ~~ . * i "- i 



4ir 4?r n + 1 



The constant term A /4ir corresponds to a magnetic shell of uniform strength 

 over S, which gives constant potential AQ at all internal points, and zero at all 

 external points. It corresponds to no system of electric currents. 



Of a certain function called the Associated Function. 



11. If S be any closed surface, and if P, Q, R be the components of a vector, such 

 that dP/dx + dQ/dy + dR/dz = at all points within S, it follows that 



Therefore there exists a determinate function, t/, of x, y, and 2, which satisfies the 

 conditions difi/dv = IP + mQ + 7iR at all points on S, and V 2 i/ = at all points 

 within S. 



I shall call this the associated function to P, Q, 11 for the surface S. 



Evidently the vector whose components are P d\l//dx, Q d\fi/dy, R d^i/dz is 

 tangential to S at every point, and forms closed curves within S. If the conditions 

 dP/dy = dQ/dx, &c., are satisfied at all points within S, then P dty/dx, Q 



MDCCCLXXXVIII. A. 2 R 



