82-83] SPHERICAL HARMONICS. 119 



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

 space ^h s in the direction (1 8 , 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_ 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, z ...h n , m! being the 

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

 for all distances r, when h ly h^...h n are diminished, and m in 

 creased, indefinitely, but so that A is finite. 



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



(6), 



,, , 

 dh 1 dh. 2 ...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 

 (h, 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 8 n . The formula (5) 

 involves 2n + 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. 



