OF TWO LEGENDRE'S COEFFICIENTS. 365 



20. We see that, when , = ! and m=l, 

 & = 2 =/*/*, + (! -ft') 1 U-/V) 



Now assume that 



From equation (3) of Section I. we have 



The coefficient of 2cosm< in <jQ n is 



/ n m M - - r/ m P r/ m P ' 



17 W^l /. 2\2/i 2\2 w J -n u * 



n 

 U ^ j 



, +i +i,, c B 



^" ( ft ' " +1 """ 



ff m p rl m ~ l P fj m+1 P 



Substituting for P-^j-^, ~d^^ and ^"^"Su^ 1 " in terms of 



/-/ p (j m p 



'_ -L n+i ( [[__n-i_ 



^ m d/t* 



cZ'"P ' 

 and similarly for frj- , &c. by means of equations (l), (5) and (6) given 



Cty4, 



above. Hence the coefficient of 2 cos m<^ in qQ n becomes 



i s a \?/ir / \/ ^d m p n+l 

 i (1 ~^ (1 ~ Mi) " {2 L- (n - m)(w - m+1) ^/^ 



rf'"-P ~l 

 )^fi 



