126 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



series we can express both these finite solutions by the single 

 formula 



----- 4 



2 . 4 . (2n - 1) (2n - 3) 



On comparison with Art. 86 (1) we find that 



.(4). 



That this is a solution of (1) may of course be verified indepen 

 dently. 



Collecting our results we learn that a surface-harmonic which 

 is finite over the unit sphere is necessarily of integral order, and is 

 further expressible, if n denote the order, in the form 



S n = A P n O) + 2 S S ^(A S cos 50) + B s sin sa&amp;gt;) T n * (/a)... (6), 



containing 2n + 1 arbitrary constants. The terms of this involving 

 &amp;lt;w are called tesseral harmonics, with the exception of the last 

 two, which are given by the formula 



(1 fj?y* n (A n cos no) + B n sin nco), 



and are called sectorial harmonics ; the names being suggested 

 by the forms of the compartments into which the unit sphere is 

 divided by the nodal lines 8 n = 0. 



The formula for the tesseral harmonic of rank s may be 

 obtained otherwise from the general expression (6) of Art. 83 

 by making n s out of the n poles of the harmonic coincide at 

 the point 6 = of the sphere, and distributing the remaining s 

 poles evenly round the equatorial circle Q = \ir. 



The remaining solution of (1), in the case of n integral may be 

 put in the form 



S n = (A s cos so) + B 8 sin sco) U n s (/i) ............ (7), 



where Z7 W - (/,) = (1 - ^^ .................. (8)*. 



This is sometimes called a tesseral harmonic of the second kind. 



* A table of the functions Q n (/j.), U n * (/*), for various values of n and s, has been 

 given by Bryan, Proc. Camb. Phil. Soc., t. vi., p. 297. 



