Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/178

Since the first derivatives of $V$ vanish at a point of equilibrium, $H_{1}=0$, if $P$ be a point of equilibrium.

Let $H_{i}$ be the first function which does not vanish, then close to the point $P$ we may neglect all functions of higher degrees as compared with $H_{i}$.

Now

$H_{i}=0$

is the equation of a cone of the degree $i$, and this cone is the cone of closest contact with the equipotential surface at $P$.

It appears, therefore, that the equipotential surface passing through $P$ has, at that point, a conical point touched by a cone of the second or of a higher degree.

If the point $P$ is not on a line of equilibrium this cone does not intersect itself, but consists of $i$ sheets or some smaller number.

If the nodal line intersects itself, then the point $P$ is on a line of equilibrium, and the equipotential surface through $P$ cuts itself in that line.

If there are intersections of the nodal line not on opposite points of the sphere, then $P$ is at the intersection of three or more lines of equilibrium. For the equipotential surface through $P$ must cut itself in each line of equilibrium.

115.] If two sheets of the same equipotential surface intersect, they must intersect at right angles.

For let the tangent to the line of intersection be taken as the axis of $z$, then ${\tfrac {d^{2}V}{dz^{2}}}=0$. Also let the axis of $x$ be a tangent to one of the sheets, then ${\tfrac {d^{2}V}{dx^{2}}}=0$. It follows from this, by Laplace’s equation, that ${\frac {d^{2}V}{dy^{2}}}=0$, or the axis of $y$ is a tangent to the other sheet.

This investigation assumes that $H_{2}$ is finite. If $H_{2}$ vanishes, let the tangent to the line of intersection be taken as the axis of $z$, and let $x=r\cos \theta$, and $y=r\sin \theta$, then, since

${\frac {d^{2}V}{dz^{2}}}=0,\ {\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}=0;$

or

${\frac {d^{2}V}{dr^{2}}}+{\frac {1}{r}}{\frac {dV}{dr}}+{\frac {1}{r^{2}}}{\frac {d^{2}V}{d\theta ^{2}}}=0;$

the solution of which equation in ascending powers of $r$ is

$V=V_{0}+A_{1}r\cos(\theta +\alpha )+A_{2}r^{2}\cos(2\theta +\alpha _{2})+\mathrm {etc} .+A_{i}r^{i}\cos(i\theta +\alpha _{i}).$