250 USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 

 rm 



Hence, employing I P n dp, m to express the mih integral of P n with regard 

 to p., we get by successive substitution 



Cm rl m P 



-\)...(n-m + -2)(n-m+l)\ P n dp.' = ( 1 - p?) m ^^ . 



10. Let 



7 Om 



Then ^ + (n - m +l)(n + m) S~- ' = 0. 



Putting m + 1 for m we have 



Now 



therefore ( 1 - /t s ) - + 2 



fl 



and T 



dp, 



or 



-~ r ,- 



1 p- (ILL 



Multiplying by (1 /t 2 )"'" we get 



the differential equation for /S m . 



/. 

 11. From Art. 9 it appears that S m and P n dp. in differ by merely 



/- 

 a constant multiplier, so that the differential equation for I P n dp." 1 is of 



the same form as the differential equation for S m , hence we get 



