USE OF LEGENDRE'S POLYNOMIALS. 



If M =f(aa'x) denotes the coefficient of mutual iuduction of the 

 two mean circles, we shall have 



M = - nn' 

 6 



f(a + c, a\ x} +f(a - c, a', x) 

 +f(a, a! + c\ x) +/(*, a' - c' , x) 

 +/(, a, x + b) +f(a, a', x-b] 

 -{-/(a, a', x + b') +/(a, a', x - b') 

 - 2/(a, a', x) 



We have only then to calculate the nine values of ^, which 

 correspond to nine values of the function/! 



766. USE OF LEGENDRE'S POLYNOMIALS. Let us suppose that 

 a system is symmetrical about an axis, and that the potential U of 

 this system at a point P on the axis at a distance x from an arbitrary 

 origin O, has been expressed by a converging series of the form 



where A , A 1 ..... , B , B x ..... are constant coefficients. If we can 

 pass from the point P to a point V outside the axis without 

 meeting acting masses, the potential U of the system at the point V 

 is represented by what is also the convergent series, 



(23) U = A + 



in which r denotes the distance OP, and X 1} X 2 . . . . , X n the 

 polynomials of Legendre (377) or the harmonic functions of 

 i, 2 . . . n order of the angle 0, which the right line OP makes with the 

 axis of symmetry. 

 If we put 



/x = cos (9, 

 from which 



and consider the polynomials X as functions of /*, Laplace's equation 

 gives the general condition 



K 2 



