206 
PROFESSOK A. SCHUSTER OX SOHE DEFINITE INTEGRALS, 
equations to determine 9 ' + 1 coefficients. The solution is conducted according to 
well-known rules, making use of the j^roposition that, if p and s are integer numbers 
smaller than q, 
2.S7r 
1 , p-TT SIT , pOTT 
■k + COS — cos-h cos — cos - 
q q q q 
+ . 
I (? — (? — I)s7r 
d- cos -cos -- - 
2 2 
+ ^ cos p-n cos HTT. 
If p and 5 are equal to each other the sum on the right-hand side is equal to \q. 
The factor ^ of the first and last term should be noted. If we designate by 
Kq, k^, k.^ ■. . . Kg, the value of k for 6—0, and at the successive circles of latitude, it 
follows tliat 
Q 
'.7 = T (< + /c,! cos p-rr) + S < cos spir^q. 
The condition under which the coefficients are determined is that the coefficients of 
the Fourier series higher than 0 “ are zero. It will he noted that, even if we suppose 
some of the lower coefficients to vanisli, the ecjuations will still give thof<c values for 
the remaining coefficients which, according to the method of least squares, fit in best 
with the assumed values of k, but that in the calcidations half weiolit onlv is olven 
to the values at the two poles. 
[The above method of obtaining the coefficients a^, which is identical with that 
in common use, when the range of 6 is 27r, is convenient whenever the function to be 
analysed is known at eveiy point of the sphere, so that the coefficients k and K may 
be determined for a siifficient number of equidistant circles of latitude. But other 
methods are available, and hence the process of obtaining the coefficients of the 
series of spherical harmonics- which I am endeavouring to explain, is not restricted to 
cases wliere the original function is known everywhere ; provided it is continuous, 
as wmll as its derivatives over the surface of the s})here. Graphical interpolation 
may be employed to determine the function at unknown points with sufficient 
accuracv, and there are several wood mechanical devices m existence, bv means of 
which at any rate the lowest and most important coefficients of the Fourier series 
may be found. It will, in some cases, materially lielj) tliis process of interpolation if 
it is remembered tliat continuity at the pole involves the vanishing of all the values 
of K- and K except — August 2, 1902.] 
Having calculated the coefficients, the reduction to spherical harmonics is made, by 
substitution of 
cos po = -f + . . . A;_iQ;-i + . . . A:Q:i + . . . , 
where the values of Al mav be calculated and tabulated once for all. Bv the 
theorem of § 8 , all coefficients vani.sh up to and inclusive of AA 2 if o- be odd, so that. 
