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

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Since the first derivatives of vanish at a point of equilibrium, , if be a point of equilibrium.

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

Now

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

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

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

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

If there are intersections of the nodal line not on opposite points of the sphere, then is at the intersection of three or more lines of equilibrium. For the equipotential surface through 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 , then . Also let the axis of be a tangent to one of the sheets, then . It follows from this, by Laplace’s equation, that , or the axis of is a tangent to the other sheet.

This investigation assumes that is finite. If vanishes, let the tangent to the line of intersection be taken as the axis of , and let , and , then, since

or

the solution of which equation in ascending powers of is