1902.] 



On Spherical Harmonic Analysis. 



97 



•• On some Definite Integrals and a New Method of reducing a 

 Function of Spherical Co-ordinates to a Series of Spherical 

 Harmonics." By Akthur Schuster, F.R.S. Received May 

 30,— Read June 5, 1902. 



(Abstract.) 



The expansion of a function f(0) of an angle 6 varying between 

 and 7r in terms of a series proceeding by the sines of the multiples of 

 depends on the fundamental theorem, 



I sin p0 sin qB dO = 0, 



o 



where p and q are integer numbers not equal to each other. Simi- 

 larly if P n denotes the zonal harmonic of degree n, /x = cos 6, and 



Ql = sin^lf^, 

 dp? 



the expansion of a function of in terms of a series of the functions 

 depends on the corresponding theorem, 



where i and n are two integer numbers not equal to each other. In 

 many practical applications a continuous function is given by means 

 of its numerical values at certain points, e.g., for equidistant values 

 of 6. 



Such cases present no difficulty when Fourier's analysis is to be 

 employed, because there is in that case a summation theorem exactly 

 corresponding to the above integration theorem. If 6 be replaced by 

 p-Tr/n, where p takes successively the values 1, 2, 3 . . . , the equa- 

 tion 



p — n— i 



S sin (ppirin) sin (pepsin) = 



p = 



will hold true. This allows us to determine the coefficients in the 

 case of problems in which discontinuous values of the function at 

 equidistant points are known (e.g., hourly readings of temperature or 

 barometric pressure). If we assume that all Fourier coefficients beyond 

 the ?ith vanish, n equations are obtained, each of which only contains 

 one of the unknown quantities. 



If it is desired to expand a function in terms of cosines, a slight 

 VOL. LXXI. i 



