162 SPHERICAL HARMONICS. [ I 3 I
into i+l in the assumed expression, equation (19), for V { . Hence the assumed form of 7J", in equation (19), if true for any value of i, is true for the next higher value.
To find the value of A Ls , put s = in equation (22), and we find
4 + i.o = ^^ 4.o ; (24)
-f- i and therefore, since A 1 is unity,
I2t
(25)
na site na nke a anyị na-enweta, site nhata (22), n'ihi na n'ozuzu uru nke
ọnụọgụgụ 12-2s
na n'ikpeazụ, uru nke okwu trigonometric maka T bụ
Nke a bụ okwu kachasị n'ozuzu maka elu okirikiri- harmonic nke ogo i. Ọ bụrụ na m atụ aka na sphere ka enyere, mgbe ahụ, ọ bụrụ na ọ bụla A na-ewere isi ihe P ọzọ na gburugburu, uru Y i maka isi ihe P bụ ọrụ nke i anya nke P site na i point, na nke i (i 1) anya nke i na-atụ aka na ibe ya. Ndị a ka m na-atụ nwere ike na-akpọ okporo osisi nke spherical harmonic. Osisi ọ bụla enwere ike ịkọwa ya site na nhazi akụkụ akụkụ abụọ, nke mere na okirikiri harmonic nke ogo m nwere 2i nọọrọ onwe ya, naanị ya oge, m i9
131.] The theory of spherical harmonics* was first given by Laplace in the third book of his Mecanique Celeste. The harmonics themselves are therefore often called Laplace s Coefficients.
They have generally been expressed in terms of the ordinary spherical coordinates and 0, and contain 2i+l arbitrary con stants. Gauss appears* to have had the idea of the harmonic being determined by the position of its poles, but I have not met with any development of this idea.
In numerical investigations I have often been perplexed on ac count of the apparent want of definiteness of the idea of a Laplace s Coefficient or spherical harmonic. By conceiving it as derived by
the successive differentiation of with respect to i axes, and as expressed in terms of the positions of its i poles on a sphere, I
- Gauss. Werlse, bd.v. s. 361.
