248 PROFESSOR A. G. GREENHILL ON THE 



where 



D 5 = (1 2m) m (1 m) 2 , 



and N 5 is obtained from D 5 by a change of m into 1 m, and a into a. 



s ( w ) ~ s ( 6v ) = 7- 



D 6 = 2 (1 - m) (1 - 2m) a m (1 m) 2 , 



N G = (1 2m + 2m 2 ) (1 2m) a m (1 m) (1 3m + 3m 2 ) ; 



/ \ / \ 2 /N 7 \ 2 /, , 



,()-, (7*)=-- -=J (16), 



a m\D 7 



D 7 = (1 2m) 3 a 3 - m(l - m)(l - 2m) (1 3m) a m 4 (l m) 2 , 

 and N 7 is obtained by a change of m into 1 m and a into a ; 



\2 



D 8 -= {(1 _ 2m) a m(l -- m)} {(1 -- 2m) a - w(l - m)(l 2m + 2m 2 )}, 



N 8 = (1 - 2m) 3 a 3 - m(l m)(l - 2m) 3 (1 + 2m 2m 2 ) a 2 



+ 4m ;! (l - ///) 3 (1 - 2m) a - m 4 (l - m) 4 ; 



D 9 = (1 -- 2m) s (3 - Gm + 4m-) a 3 2m (I m)(l -- 2m) 2 (3 7m + 5m 2 ) a 2 



+ m 8 (l -- m)-(l - 2m) (4 - 10m + 7m 2 ) a m 3 (l m) 6 , 



and a change of a into a and m into 1 m gives N 9 ; 



D io = {(1 - 2m) a - m(l - m) 2 } {(1 - 2m) a - m 2 (l - m)} 



{(1 - 2m)* a 2 - m (1 - m) (1 - 2m) (1 - Gm + 6m 2 ) a - m 3 (1 m 3 )}, 



N 10 = (1 - 2m) c a 4 - 2m (1 - m) (1 - 2m) 3 (2 - 7m + 7m 8 ) a 3 



+ 2m 2 (1 - m) 3 (1 - 2m) 2 (3 - llm + 13m 2 4m 3 + 2m 4 ) a 2 

 m 3 (1 m) 3 (1 2m) (4 17m + 27m 2 20m 3 + 10m 4 ) a 

 + m 4 (1 m) 4 (1 5m + 10m 2 - 10m 3 + 5m 4 ). 

 A simplification is effected by putting 



(1 2m) = m (1 m) ft, and 1 2m = p (20), 



