Potential of a Symmetrical System. 239 



where 



A„=iO+i) f +1 /M p »o*>4*' 



Here the necessary conditions for the expansion are satisfied, 

 and we find 



i. e. A r = (2r + 1) fiP r (fj)dfi, (r even) 



= 0. (r odd) 



Now 



(2r+l)/*P,M = (r + lJPr+iM +rPr-iW, 

 and so, if r=2n, 



A 2)l = f 1 [(2/i + l)P 2n+1 (/*) + 2nP 2n _ 1 (/*)]^. 



It is easy to prove that 



It- n-\ 



and so 



1 P2»-l(/*)^A t = c »« 



This gives 



Aa,= (2n + l)c B+ i+2nc ft . 



But from the definition of the c's it follows that 



c„+i _ \~ n = __ 2>i~l 

 c„ ' n+1 2(» + l)' 



and thus 



A 2 „=(c n -c„ + i) + [(2/j — l>»+2(/i + l)f H+ i], 



Also 



A =i I /M^=J ^=i=c — ci. 



