MR. S. H. BURBURY ON THE INDUCTION OP ELECTRIC 



40. If any one of a series of similar shells Avith similar currents be self-inductive, 

 every one of the shells is self-inductive. 



Tf all the shells within S be filled with similar currents, as above defined, the 

 components of current per unit of area must be 



dS d^> dS d$> 

 dz dy dy dz 



&C. = &c. 



Let us now find a condition that, if a system of currents of the type < be generated 

 in the outer shell, any inner shell of the series, if a conductor and exposed to the 

 influence of the outer, shall have the corresponding system of currents of the same 

 type excited in it. 



At any point on the outer shell S, since the shell is self-inductive, we have 



/_ dy\ /dSdd> dSd(f>\ 



F -j*) = oK = <r(-r:r J~ 



\ dxj \dz dy dy dz/ 



where F is the component of vector potential of the currents in the outer shell S, 

 and x the associated function. Now, v 2 (F d\]dx) = at all points within S. If, 

 therefore, v 2 = 0, u being a harmonic of positive degree, at all points within S, it 

 follows that 



Ac, d X\ fdSdd> dSdd>\ 



K F - -j* = <ru = o- ( -f- - H 



\ ax i \dz dy dy dz/ 



at all points within S. The same is true of G d\ldy if - ----- -, - is a solid 



y dx dz dz dx 



harmonic of positive degree, &c. Let S be any one of the shells within S, and 

 suppose it to become a conductor. If it be self-inductive to the system of currents 

 u, v, w, it follows that this system with reversed sign, and no other, will be excited in 

 it by induction when the corresponding system is excited in the shell S. For instance, 

 in the case of the ellipsoid above treated, 



<}> = Az, 



dSdQ _ dSd$ Ay 



dz dy dy dz ""6 s 



z z "" 



and 



\dz dy dy dz 



If, therefore, the ellipsoid x 2 /a z + y z /b 2 + z 2 /c 2 = 1 be divided into similar, similarly- 

 situated, and concentric ellipsoidal shells, each shell is, as we have proved above, self- 



