Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/208

Hence

${\begin{array}{l}{\frac {d^{2}H_{i}}{dx^{2}}}+{\frac {d^{2}H_{i}}{dy^{2}}}+{\frac {d^{2}H_{i}}{dz^{2}}}=(2i+1)(2i+2)r^{2i-1}V_{i}\\\\+2(2i+1)r^{2i-1}\left(x{\frac {dV_{i}}{dx}}+y{\frac {dV_{i}}{dy}}+z{\frac {dV_{i}}{dz}}\right)+r^{2i+1}\left({\frac {d^{2}V_{i}}{dx^{2}}}+{\frac {d^{2}V_{i}}{dy^{2}}}+{\frac {d^{2}V_{i}}{dz^{2}}}\right).\end{array}}$

(44)

Now, since $V_{i}$ is a homogeneous function of negative degree $i+1$,

$x{\frac {dV_{i}}{dx}}+y{\frac {dV_{i}}{dy}}+z{\frac {dV_{i}}{dz}}=-(i+1)V_{i}.$

(45)

The first two terms therefore of the right hand member of equation (44) destroy each other, and, since $V_{i}$ satisfies Laplace’s equation, the third term is zero, so that $H_{i}$ also satisfies Laplace’s equation, and is therefore a solid harmonic of degree $i$.

We shall next shew that the value of $H_{i}$ thus derived from $V_{i}$ is of the most general form.

A homogeneous function of $x,y,z$ of degree $i$ contains

${\frac {1}{2}}(i+1)(i+2)$

terms. But

$-\nabla ^{2}H_{i}={\frac {d^{2}H_{i}}{dx^{2}}}+{\frac {d^{2}H_{i}}{dy^{2}}}+{\frac {d^{2}H_{i}}{dz^{2}}}$

is a homogeneous function of degree $i-2$, and therefore contains ${\tfrac {1}{2}}i(i-1)$ terms, and the condition $\nabla ^{2}H_{i}=0$ requires that each of these must vanish. There are therefore ${\tfrac {1}{2}}i(i-1)$ equations between the coefficients of the ${\tfrac {1}{2}}(i+1)(i+2)$ terms of the homogeneous function, leaving $2i+1$ independent constants in the most general form of $H_{i}$.

But we have seen that $V_{i}$ has $2i+1$ independent constants, therefore the value of $H_{i}$ is of the most general form.

Application of Solid Harmonics to the Theory of Electrified Spheres.

134.] The function $V_{i}$ satisfies the condition of vanishing at infinity, but does not satisfy the condition of being everywhere finite, for it becomes infinite at the origin.

The function $H_{i}$ satisfies the condition of being finite and continuous at finite distances from the origin, but does not satisfy the condition of vanishing at an infinite distance.

But if we determine a closed surface from the equation

$V_{i}=H_{i},$

(46)

and make $H_{i}$ the potential function within the closed surface and