82-83] SPHERICAL HARMONICS. H9 



derive a system of higher order by first displacing it through a 

 space %hg in the direction (1 S) m s , n s \ and then superposing the 

 reversed system, supposed displaced from its original position 

 through a space ^h s in the opposite direction. Thus, beginning 

 with the case of a simple source at the origin, a first application 

 of the above process gives us two sources 0+, 0_ equidistant from 

 the origin, in opposite directions. The same process applied to the 

 system + , 0_ gives us four sources + + , 0_ + , 0+_, __ at the 

 corners of a parallelogram. The next step gives us eight sources at 

 the corners of a parallelepiped, and so on. The velocity-potential, 

 at a distance, due to an arrangement of 2 n sources obtained in 

 this way, will be given by (5), where A = m'hji 2 ...h n) m' being the 

 strength of the original source at 0. The formula becomes exact, 

 for all distances r, when h lt h. 2 ,...h n are diminished, and m in- 

 creased, indefinitely, but so that A is finite. 



The surface -harmonic corresponding to (5) is given by 



fin 1 



(6), 



. . .dh n r 

 and the complementary solid harmonic by 



(7). 



By the method of ' inversion *,' applied to the above configura- 

 tion of sources, it may be shewn that the solid harmonic (7) of 

 positive degree n may be regarded as the velocity-potential due 

 to a certain arrangement of 2 n simple sources at infinity. 



The lines drawn from the origin in the various directions 

 (l s , m s , n s ) are called the 'axes' of the solid harmonic (5) or (7), 

 and the points in which these lines meet the unit sphere are 

 called the 'poles' of the surface harmonic S n . The formula (5) 

 involves 2^+1 arbitrary constants, viz. the angular co-ordinates 

 (two for each) of the n poles, and the factor A. It can be shewn 

 that this expression is equivalent to the most general form of 

 surface-harmonic which is of integral order n and finite over the 

 unit sphere f. 



* Explained by Thomson and Tait, Natural Philosophy, Art. 515. 

 t Sylvester, Phil. Mag., Oct. 1876. 



