WITH APPLICATION TO SPHERICAL HAH.MONIC ANALYSIS. 
205 
to M}1 The last value so obtained, and some of the intermediate ones, were calculated 
independently from the series defining M, numerical errors being easily detected in 
this wav. The values of Sh Uy were dealt with in a similar manner. As a 
final check, the values of Sfl and were obtained from and by means of the 
second and fourth of the equations (L). 
§ 11. Application of the Previous Results to the Expression of Functions hy Means 
of a Series of Spherical Harmonics. 
The results obtained in this investigation lead to a new method of calculating the 
coefficients of the series of spherical harmonics which represent a function F of 
spherical co-ordinates. If the values of this function are given for all points of a 
sphere, the coefficients of the term involving Tfi is known to depend solely on the 
integral j’FTfi r/8 taken over the surface of the sphere. This theoretically perfect 
proceeding has to Ije modified in practice when the values of F are known only at 
definite points, so that the intermediate positions liave to be evaluated hy a process 
of interpolation on the supposition tliat F is continuous everywliere. 
All methods which have been applied so far, sufier from tlie serious inconvenience 
that the method of least squares is applied in such a way as to make the value of 
the coefficients of lower degrees depend on the number of terms whicli are taken 
into account. That is to say, the coefficients are not independent of each other as 
they ought to be. 
A. method suggested by F. E. Neumann, Avhich is free from this defect, introduces 
other complications, and has never, as far as I kno'w, ])een applied in practice. 
The theorem of § 8 oilers a simple solution of the practical difficulties, and reduces 
the whole problem to an expansion Iry means of Fourier’s analysis, which can be 
carried out either hv the well-knoAvn process of calculation, or by mechanical means. 
Let F be expressed, in the first place, for different circles of latitude as a series 
proceeding by cosines and sines of the longitude and of its multiples. This first 
step is common to all methods. 
The result may l)e expre.ssed symbolicallv bv 
F = /<" -F K cos (f) + K' cos A- • ■ ■ + sin f -f K- sin '2<j) + . . . , 
wliere P, S, K/, K- . . . :ire functions of the colatitude. If their values are known 
at ij — 1 equidistant circles of latitude, and at the poles, we may detei'mine the 
coefficients upland uf Avhich satisfy the equations 
k"' = 0(1 a I cos 0 -|- aZ cos 2$ E • • • < C'ls p9 f . . . a^^^ cos qO, 
= E + E cos 0 -h «.T cos 20 E . . . rf, cos 2j9 E • • • “I 
If we give to 9 the successive Auilues 0, vlq, 2Trlq . , . qrr/q, we shall have ^ + 1 
