OF TWO LEGENDRE'S COEFFICIENTS. 367 



And the coefficient of 2cosm0 in nQ n _^ is 



m m ( lmvn~\\\ rl m P fJ m P' 



(^ -u. a v n -iL-Y 2 J -n { ' ""' =- 



f*j ii M i n (H+m_i)! d/t 4c 



Hence adding these last results and dividing by (n + l), the coefficient 

 of 2cosm< in Q H+l is 



, , n n1 n+1 



l)\ ( *' M dp." dtf 



Hence the same law holds good for Q n+1 , and since the expression 

 assumed is evidently true when n = and when n=l, it is true generally. 



21. (The same proof is applicable to the term independent of <, 

 i.e. when m = 0.) 



The term independent of $ in Q n q is 



Now u-P - ~ - P 



* n n+l dp 



and (2n + 1 ) pP n = (n + 1 ) P 



.-. (2n + 1)* ^P n P n ' = [(n + 1) P B+1 + nP^ [_(n + 1) P' n+ , + nP' 



therefore the term independent of (^ in (2n + l) 2 ^ 9l r/ is 

 [(w + 1 ) P B+1 + nP..,] [(n + 1 ) P' K+1 + 



= (2w+ 

 therefore the term independent of <f> in 



(n4l)Q )S+1 or l&n+^Qa-nQ^ is 

 Hence the first term of Q n+1 is P +l P' n+l , and the law is true generally. 



