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IV. On some Definite Integrals^ and a Xew 2[efhod of Reducing a, Function of 
Sphericcd Co-ordinates to a Series of Spherical Harmonics. 
Bij Arthur .Schuster, F,R.S. 
Koccived May 30 ,—-FvCud June 5 , 1002 . 
^ 1. Introd'ictorg. 
The following investigation deals with some detinite integrals which are useful when 
it is desired to express a function of two angular wiriables hy means of a series of 
spherical surface harmonics. An important theorem concerjiing these integrals leads 
to a method which considerably reduces the- arithmetical labour involved in the 
reductions, and secures in practice the advantage of obtaining the numerical values 
of the coellicients of lower degrees independently of those of higlier degrees. 
The zonal harmonic of degree n is denoted by and defined as usual by 
1 
7^ (i^'- 1)A 
2 " 1-2 ... ■ 
The tesseral harmonic of degree n and type cr is denoted by 
where 
= Q;i(.Vl cos a-</> -i- ir sin 
Q: = sin-^ 0 
i/At, 
In these equations 0 represents an angle measured on a .spliere from a })olnt as pole 
and fi = cos 0. The longitude measured from some standaixl meridian is denoted 
by p 
The name given by Heine, and translated by Tohhunter as “ Associated Functions 
of the First Kind,” is too cumbersome for use, and I propose to call these functions 
“ Tesseral Functions.” The tesseral function is converted into a tesseral harmonic by 
the factor cos cr0 or sin cr</). The name may not perhaps appear to be appropriate, 
because it is only the factor wliicli gives the function its “tesseral” character, Imt it 
IS short and suggests at once the function it denotes. 
The present investigation deals in great part with the definite integrals, taken over 
the surface of a sphere of unit- radius, <h‘ the product of two tesseral functions and of 
(324.) 18.12.02 
