126 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



series we can express both these finite solutions by the single 

 formula 



2.4.(2n-l)(2n-3) 



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



2 (2n 1) 



^- s - 4 -.J (4). 



(5). 



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 Q P n (fi) + 2;::(^l s cos sco + B 8 sin sco) T n s (p) . . . (6), 



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

 a) are called ' tesseral ' harmonics, with the exception of the last 

 two, which are given by the formula 



(1 - ffi n (A n cos nco -f 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 S 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 = of the sphere, and distributing the remaining s 

 poles evenly round the equatorial circle = ^7r. 



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

 put in the form 



n s (fj,) ............ (7), 



where U n * fa) = (1 - ^ .................. (8)-. 



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



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

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



