Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/217

at equal intervals ${\tfrac {\pi }{\sigma }}$ round the equator, is the type $2\sigma$, if one of the poles is at the origin of longitude, or the type $2\sigma +1$ if it is at longitude ${\tfrac {\pi }{2\sigma }}$.

143.] It appears from equation (60) that it is always possible to express a harmonic as the sum of a system of zonal harmonics of the same degree, having their poles distributed over the surface of the sphere. The simplification of this system, however, does not appear easy. I have however, for the sake of exhibiting to the eye some of the features of spherical harmonics, calculated the zonal harmonics of the third and fourth degrees, and drawn, by the method already described for the addition of functions, the equipotential lines on the sphere for harmonics which are the sums of two zonal harmonics. See Figures VI to IX at the end of this volume.

Fig. VI represents the sum of two zonal harmonics of the third degree whose axes are inclined 120° in the plane of the paper, and the sum is the harmonic of the second type in which $\sigma =1$, the axis being perpendicular to the paper.

Fig. VI

In Fig. VII the harmonic is also of the third degree, but the axes of the zonal harmonics of which it is the sum are inclined 90°, and the result is not of any type of the symmetrical system. One of the nodal lines is a great circle, but the other two which are intersected by it are not circles.

Fig. VII

Fig. VIII represents the difference of two zonal harmonics of the fourth degree whose axes are at right angles. The result is a tesseral harmonic for which $i=4$, $\sigma =2$.

Fig. VIII

Fig. IX represents the sum of the same zonal harmonics. The result gives some notion of one type of the more general harmonic of the fourth degree. In this type the nodal line on the sphere consists of six ovals not intersecting each other. Within these ovals the harmonic is positive, and in the sextuply connected part of the spherical surface which lies outside the ovals, the harmonic is negative.

Fig. IX

All these figures are orthogonal projections of the spherical surface.

I have also drawn in Fig. V a plane section through the axis of a sphere, to shew the equipotential surfaces and lines of force due to a spherical surface electrified according to the values of a spherical harmonic of the first degree.

Fig. V