212 Methods for the Solution of Special Problems [CH. vin 



(i) The rational integral harmonics of degree n are (2n + 1) in number, 

 and may all be derived from the harmonic - by differentiation. 



(ii) Any function of position on a spherical surface, which satisfies the 

 conditions which obtain in a physical problem, can be 

 expanded as a series of rational integral harmonics, 

 and this can be done only in one way. 



243. Before considering these harmonics in detail, 

 we may try to form some idea of the physical concep- 

 tions which lead to them most directly. 



The function - is the potential of a unit charge at 



the origin. If, as in 64, we consider two charges 

 e at points 0', 0" at equal small distances a, a 

 from the origin along the axis of x, we obtain as the 

 potential at P, 



e e 







O'P 0"P OP" OP' 



V- 



If we take e . PP" = 1, we have a doublet of strength 1 parallel to the 

 axis of x, and the potential at P is =- (-). In fact this potential is exactly 



OX \T J 



the same as - already found in 64. 



Thus the three harmonics of order 1 obtained by dividing the rational 



a /IN a /i\ a /i\ 



integral harmonics of order 1 by r 3 , namely 5 I - 1 o~~ - ] -5- ( ~ 1* are simply 



J dx \rj dy \rj dz \rj 



the potentials of three doublets each of unit strength, parallel to the negative 

 axes of x, y, z respectively. 



If in fig. 73 we replace the charge e at 0' by a doublet of strength e 

 parallel to the negative axis of x, and the charge e at 0" by a doublet 

 of strength e parallel to the negative axis of x t we obtain a potential 



a 2 



If instead of the doublets being parallel to the axis of x y we take them 

 parallel to the axis of y, we obtain a potential 



a 2 /r 



'dx'dy 



So we can go on indefinitely, for on differentiating the potential of 

 a system with respect to x we get the potential of a system obtained 



