256-259] Spherical Harmonics 



Similarly the potential at any external point P is 



221 



These potentials are obviously solutions of Laplace's equation, and it is 

 easy to verify that they correspond to the given surface density, for 



outside 



inside 



This gives us the fundamental property of harmonics, on which their 

 application to potential-problems depends : A distribution of surface density 

 S n on a sphere gives rise to a potential which at every point is proportional 



toS n . 



258. The density of the most general surface distribution can, by the 

 theorem of 240, be expressed as a sum of surface harmonics, say 



in which $ is of course simply a constant. The potential, by the results of 

 the last section, is 



F= 47ra -U + (-} + f-V + ... at an internal point ...(174), 

 ( o \aj o \aj J 



an external point ...(175). 



= 4 



~Y 

 o \r J 



EXAMPLES OF THE USE OF HARMONICS. 



I. Potential of spherical cap and circular ring. 



259. As a first example, let us find the potential of a spherical cap 

 of angle a i.e. the surface cut from a sphere by 

 a right circular cone of semivertical angle a 

 electrified to a uniform surface density cr Q . 



We can regard this as a complete sphere 

 electrified to surface density cr, where 



cr = o- from 6 = to 6 = a, 

 o- = from 6 = 0. to 6 TT. 

 The value of a being symmetrical about the 

 axis = 0, let us assume for the value of cr ex- 

 panded in harmonics 



cos 6) + a 2 ^(cos 9) 



Fm. 76. 



