Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/213

be satisfied in order that it may be conjugate to the $2i$ preceding harmonics.

Hence a system of $2i+1$ harmonics of degree $i$ may be constructed, each of which is conjugate to all the rest. Any other harmonic of the same degree may be expressed as the sum of this system of conjugate harmonics each multiplied by a coefficient.

The system described in Art. 132, consisting of $2i+1$ harmonics symmetrical about a single axis, of which the first is zonal, the next $i-1$ pairs tesseral, and the last pair sectorial, is a particular case of a system of $2i+1$ harmonics, all of which are conjugate to each other. Sir W. Thomson has shewn how to express the conditions that $2i+1$ perfectly general harmonics, each of which, however, is expressed as a linear function of the $2i+1$ harmonics of this symmetrical system, may be conjugate to each other. These conditions consist of $i(2i+1)$ linear equations connecting the $(2i+1)^{2}$ coefficients which enter into the expressions of the general harmonics in terms of the symmetrical harmonics.

Professor Clifford has also shewn how to form a conjugate system of $2i+1$ sectorial harmonics having different poles.

Both these results were communicated to the British Association in 1871.

139.] If we take for $Y_{j}$ the zonal harmonic $Q_{j}$, we obtain a remarkable form of equation (57).

In this case all the axes of the second system coincide with each other.

The cosines of the form $\mu _{ij}$, will assume the form $\lambda$ where $\lambda$ is the cosine of the angle between the common axis of $Q_{j}$ and an axis of the first system.

The cosines of the form $\mu _{ij}$ will all become equal to unity.

The number of combinations of $s$ symbols, each of which is distinguished by two out of $i$ suffixes, no suffix being repeated, is

$N={\frac {|{\underline {i}}}{2^{s}|{\underline {s}}\ |{\underline {i-2s}}}};$

(58)

and when each combination is equal to unity this number represents the sum of the products of the cosines $\mu _{jj}$, or $\sum \left(\mu _{jj}^{s}\right)$.

The number of permutations of the remaining $i-2s$ symbols of the second set of axes taken all together is $|{\underline {i-2s}}$. Hence

$\sum \left(\mu _{ij}^{i+2s}\right)=|{\underline {i-2s}}\sum \lambda ^{i-2s}.$

Equation (57) therefore becomes, when $Y_{j}$ is the zonal harmonic,