In this expansion the coefficient of is unity, and all the other terms involve . Hence at the pole, where and , .
It is shewn in treatises on Laplace’s Coefficients that is the coefficient of in the expansion of .
The other harmonics of the symmetrical system are most conveniently obtained by the use of the imaginary coordinates given by Thomson and Tait, Natural Philosophy, vol. i. p. 148,
The operation of differentiating with respect to a axes in succession, whose directions make angles with each other in the plane of the equator, may then be written
The surface harmonic of degree and type is found by differentiating with respect to axes, of which are at equal intervals in the plane of the equator, while the remaining coincide with that of , multiplying the result by and dividing by . Hence
where the factor 2 must be omitted when .
The quantity is a function of , the value of which is given in Thomson and Tait’s Natural Philosophy, vol. i. p. 149.
It may be derived from by the equation
where is expressed as a function of only.