30G MR. S. H. BURBUBY ON THE INDUCTION OF ELECTRIC 



and R d\j>/dz are severally zero at all points within S. For, since these conditions are 

 satisfied, there must exist a function x f x > V> an( ^ 2 8ucn that P = ty/dx, Q = dx/dy, 

 R = dx/dz at all points within S. And therefore P d^t/dx = d(x ifi/dx, &c. ; 

 and therefore d (x $)/dv = at all points on S, and V 2 (x V*) = at all points 

 within S ; and therefore x 'A = constant, and P d^dx = 0, &c., at all points 

 within S. 



12. If P, Q, R be the components of an electromotive force, and if S be a conductor, 

 then, whether the condition P d^jdx = 0, &c., be satisfied or not, they will produce 

 on S a distribution of electricity having potential />. For let 



and 



p _^ 



*1 - 7 



dx 



Similarly, 



M 



Q = Qi + Q' = ^~ + Q', 



R = RI -f- R' = -j- R ; 



then the vector or electromotive force whose components are P x> Q u R x is derived 

 from the potential $, and produces on S a distribution having potential vft. The vector 

 whose components are P', Q', R', or P d^/dx, Q d^jdy, R d\jj/dz, forms closed 

 curves within or upon S, and cannot affect the potential. 



13. Now let F , G , H be the components of vector potential due to a magnetic 

 system external to S. Let 1/ be their associated function. Let the magnetic 

 screen to the external system be formed on S. Let F, G, H be the components of 

 vector potential due to it, and let /> be their associated function. Then for the 

 two systems together we have a vector whose components are F + F, &c., with 

 (\II Q + i//) for associated function ; and, since d (F + F)/cfy = d (G + G)/dx, &c., 

 within S because of the screen, it follows that 



O + G + j (i/ + iff) = j> at all points within S. 



ay 



14. We are now in a position to consider the general problem of induction, when 

 electric conductors of any shape are in a magnetic field and the field is made to vary. 

 Electric currents are generated by induction in or on the surface of the conductors. 



