190 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



andsmceforr=0, ^ = -, ^--, 



Now multiplying and dividing each term by r n+l , we find 



(22) a- {^+^+5^+ +*+ 



where 



This is the determination of the constant A adopted by 

 Legendre, for the reason that, since by the binomial theorem, for 

 r <r, and /z.= 1, 



r r r 



it makes for every n, 



(23) p.(i)=i. 



The term P n /r n+1 is a spherical harmonic of degree (n + 1), 

 and the series (22) is convergent for r' < r. In like manner if 

 r' > r we find 



In order to find P n as a polynomial in //, we may write r/c2 as 



and develop by the binomial theorem. 



Developing the last factor 



