118 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



S n will be a function of the direction (only) in which the point 

 (x, y, z) lies with respect to the origin ; in other words, a function 

 of the position of the point in which the radius vector meets a 

 unit sphere described with the origin as centre. It is therefore 

 called a ' surface-harmonic ' of order n. 



To any solid harmonic <fr n of degree n corresponds another of 

 degree n 1, obtained by division by r m+1 ; i.e. < = r~' 2n ~ l <f) n is 

 also a solution of (1). Thus, corresponding to any surface har- 

 monic S n , we have the two solid harmonics r n S n and r~* l ~ l S n . 



83. The most important case is when n is integral, and when 

 the surface-harmonic S n is further restricted to be finite over the 

 unit sphere. In the form in which the theory (for this case) is 

 presented by Thomson and Tait, and by Maxwell*, the primary 

 solution of (1) is 



<!>-! = A IT (3). 



This represents as we have seen (Art. 56) the velocity-potential 

 due to a point-source at the origin. Since (1) is still satisfied 

 when <j) is differentiated with respect to x, y y or z, we derive a 

 solution 



d_ d^ d\l 



This is the velocity-potential of a double-source at the origin, 

 having its axis in the direction (I, m, n). The process can be 

 continued, and the general type of solid harmonic obtainable in 

 this way is 



4 tv J- / r \ 



, d , d d d 



where -==- = 1 8 -j- - + m 8 -j- + n s -j- , 



dh s dx dy dz 



1 8 > >s, n s being arbitrary direction-cosines. 



This may be regarded as the velocity-potential of a certain 

 configuration of simple sources about the origin, the dimensions 

 of this system being small compared with r. To construct this 

 system we premise that from any given system of sources we may 



* Electricity and Magnetism, c. ix. 



