Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/89

must be convergent however great r may be, we must have only negative powers of r, of the form

$B_{3}S_{i}r^{-(i+1)}.\,$

The conditions which must be satisfied by the function Ω, are: It must be (1) finite, and (2) continuous, and (3) must vanish at an infinite distance, and it must (4) everywhere satisfy Laplace's equation.

On account of (1) B₁ = 0.

On account of (2) when r = a₁,

$(A_{1}-A_{2})a_{1}^{2i+1}-B_{2}=0,\,$

(2)

and when r = a₂,

$(A_{2}-A_{3})a_{2}^{2i+1}+B_{2}-B_{3}=0.\,$

(3)

On account of (3) A₃ = 0, and the condition (4) is satisfied everywhere, since the functions are harmonic.

But, besides these, there are other conditions to be satisfied at the inner and outer surface in virtue of equation (10), Art. 427.

At the inner surface where r = a₁,

$(1+4\pi \kappa ){\frac {d\Omega _{2}}{dr}}-{\frac {d\Omega _{1}}{dr}}+4\pi \kappa {\frac {dV}{dr}}=0,$

(4)

and at the outer surface where r = a₂,

$-(1+4\pi \kappa ){\frac {d\Omega _{2}}{dr}}+{\frac {d\Omega _{3}}{dr}}-4\pi \kappa {\frac {dV}{dr}}=0.$

(5)

From these conditions we obtain the equations

$(1+4\pi \kappa )(iA_{2}a_{1}^{2i+1}-(i+1)B_{2})-iA_{1}a_{1}^{2i+1}+4\pi \kappa iC_{i}a_{1}^{2i+1}=0,$

(6)

$(1+4\pi \kappa )(iA_{2}a_{2}^{2i+1}-(i+1)B_{2})+(i+1)B_{3}+4\pi \kappa iC_{i}a_{2}^{2i+1}=0,$

(7)

and if we put

$N_{i}={\frac {1}{(1+4\pi \kappa )(2i+1)^{2}+(4\pi \kappa )^{2}i(i+1)\left(1-\left({\frac {a_{1}}{a_{2}}}\right)^{2i+1}\right)}},$

(8)

we find

$A_{1}=-(4\pi \kappa )^{2}i(i+1)\left(1-\left({\frac {a_{1}}{a_{2}}}\right)^{2i+1}\right)N_{i}C_{i},$

(9)

$A_{2}=-4\pi \kappa i\left[2i+1+4\pi \kappa (i+1)\left(1-\left({\frac {a_{1}}{a_{2}}}\right)^{2i+1}\right)\right]N_{i}C_{i},$

(10)

$B_{2}=4\pi \kappa i(2i+2)a_{1}^{2i+1}N_{i}C_{i},$

(11)

$B_{3}=4\pi \kappa i(2i+1+4\pi \kappa (i+1))(a_{2}^{2i+1}-a_{1}^{2i+1})N_{i}C_{i}.$

(12)

These quantities being substituted in the harmonic expansions give the part of the potential due to the magnetization of the shell. The quantity N_i is always positive, since 1+4πκ can never be negative. Hence A₁ is always negative, or in other words, the