259-261] 



Spherical Harmonics 



223 



and expression (178) is the only expansion in Lagrange's coefficients which 

 satisfies Laplace's equation and agrees with this expression when = 0. 



II. Uninsulated sphere in field of force. 



261. The method of harmonics enables us to find the field of force 

 produced when a conducting sphere is introduced into any permanent field 

 of force. Let us suppose first that the sphere is uninsulated. 



FIG. 77. 



Let the sphere be of radius a. Round the centre of the sphere describe 

 a slightly larger sphere of radius a', so small as not to enclose any of the 

 fixed charges by which the permanent field of force is produced. Between 

 these two spheres the potential of the field will be capable of expression in a 

 series of rational integral harmonics, say 



(179). 



The problem is to superpose on this a potential, produced by the 

 induced electrification on the sphere, which shall give a total potential 

 equal to zero over the sphere r = a. Clearly the only form possible for 

 this new potential is 



Thus the total potential between the spheres r = a and r = a' is 



