97 100] NEWTONIAN POTENTIAL FUNCTION. 189 



If we put in this V n = r n Y n we obtain 



as the differential equation satisfied by a surface harmonic 

 Yn (@, <) This is the form of Laplace's equation originally given 

 by Laplace*. If T n is the zonal harmonic P n , which is independent 

 of <p, we have 



(17) ( 



or putting cos 6 = //,, 



This is known as Legendre's Differential Equation. We shall 

 now, without considering more in detail the general surface har- 

 monic, find the general expression for the zonal harmonic. It 

 may be at once shown, by inserting for P n (/A) a power-series in fju 

 and determining the coefficients, that for integral values of n the 

 differential equation is satisfied by a polynomial in p. The form 

 of these polynomials we shall find from one of their important 

 properties. 



100. Development of Reciprocal Distance. We know 

 that l/r, the reciprocal of the distance of the point x, y, z from 

 any fixed point P, is a harmonic function of the coordinates 

 x, y, z, and although it is not a homogeneous function except when 

 the fixed point is the origin, it may always be developed in a 

 series of homogeneous functions, that is, in a series of spherical 

 harmonics. We shall now use the letter d for the distance from 

 any fixed point, reserving r for the distance from the origin. Let 

 us for convenience take the axis of z as passing through the fixed 

 point P, which lies at a distance r' from the origin, and put 

 cos (rz) = ft. Then we have 



09) \ = l> 2 + r '* ~ 2rr>]~^ = [> 2 + f + (z - r') 2 ]~ I 



Considering this as a function of z let us develop by Taylor's 

 Theorem, 



f?r\\ f(v v'\ 



-o; d -f(*- -j v/ - v~ / j ^/ r/=o - 21 



* Laplace, "Theorie des attractions des spheroides et de la figure des planetes." 

 Mem. de VAcad. de Paris. Annee 1782 (pub. 1785). 



