xx PREFACE TO PART II. 



If then, as before, we put r=l, we shall have the final equation for 

 a,, as follows : 



= f X; x m dp. + I ' Y: y m dp + (n+l) \ H: z m dp, 

 J -i J -i J -t 



where the coefficient of /3 M = 0. 



And a l (X':Y dp + f ( F: ) 2 ^ - n (n + 1 ) P (H^ ? dp] 



J -i J -i 



= | XI x m dp + i Yl y m dp -n! H: z m dp, 

 J -i J -i J -i 



where the coefficient of a n = 0. 



Hence a n and fi n are separately determined from the equations 



2a (n f- 1^ (^-^)!(^ + m)! 

 ) [l.3.5...(2w-l)J J 



= f ' jf: a m d/i + [' r y m dp + ( + 1) f ^: 2 m rf/i, 



j -i j -i j -i 



and 25 (-m)!(w + m)! 



^ n [1.3.5...(27 2 .-l)j 2 



= f 1 x: x m dp + 1 1 Y: y m dp -n( l H: Zm dp. 



J -1 J -1 J -1 



Thus generally from the values of X and Y we derive 



= (2n + 1) [ f 1 JT; ;, ^ + [' F,r y. dp] , 

 <-J -i J -i 



and from the values of Z we derive 



-n&j T (H:ydp= T jy;"^^. 

 j -i j -i 



