244 USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 



Let the indefinite integral of f(^} =./i (/A), the integral being supposed 

 to vanish when p. = 1 ; also let the indefinite integral of f t (p.) on the 

 same supposition be / (fj.) and so on, till we come to 



]/_,(/*) <?/*=/.(/*) 



Then integrating by parts we have 



= /r /, (/,) - 2rf._ (p.) + 2f 3 (p.), &c. = &c., 



= ft*" 1 / (/-O - (n ~ 1) ^- a / s M + (n- 1) (n- 2) /*-/, (/i) 4- ... 



-2 ...2. 



all the integrals on the left-hand sides of equations being supposed to vanish 

 when p. 1. 



Now put fj- = 1 in these integrals, and we have 



so that f n (iJ.) and its first ?i I differential coefficients vanish when /,= !, as 

 well as when = I. 



Hence /(//) is divisible both by (1 /A)" and by (!+/A)", and since it is 

 of 2n dimensions in /x it must be of the form 



TT ft \ d ' 1 /! \ 



Hence / (u) = c r - ( 1 ir) . 



dp.' 1 

 If c be chosen so thaty(/z) = l when //, = !, we have 



( 1 )" 1 

 therefore " 



c = 



2" ?i! 





