288 PKOFESSOR A. G. GREENHILL ON THE 



E (J = a 6 + a 3 (- 1 + + - 6a 3 + + 3a 5 + a 6 ) // 



+ 3n 3 (2 + 2a - 2a 2 - a 3 ) 6 8 - (1 + Sa + - 3a 3 ) 6 12 (49), 



F n = A + A^ + A 2 6 8 +A 3 6 12 + A,6 16 + A 6 6 20 (50), 



E n == B 5 + B 4 i* + B 3 fc 8 + B,/> 12 + B^ 16 + B 6 20 (51), 



where the A's are found in 45 (3), and the B's are derivable by the substitution (a, ). 



\ <*/ 



These expressions have been tested by putting 



F 5 = 0, b 4 = - a + a 2 + a 3 (52), 



which makes 



E 7 = a 2 (1 - a 8 ) 3 , F 7 = a (1 - a 2 ) 3 (53), 



E 9 = a-' (1 + a) 5 (1 - a) 4 , F 9 = a 2 (1 + a) 5 (1 - a) 7 (54), 



E u = a 5 (1 + ) 8 (1 - a) 7 , F,, = 4 (1 + *) 8 (1 - a) 7 (55), 



and thus verifies 



dn7/_ o 1+6 dn9/_ dnll/_ a + b 



o dn3/"" I -b' o o " ~ a-b 



42. M = 20, /= 1 ' 3 ' 7 ' 9 . 



^ 4 = - a + a 2 (- a 3 , a 3 6* = a (1 a) (1), 



C o _ + 64a*6* ,x 



- (1 + a 2 ) (1 + 2a - 6a 2 - 2a 3 + a*) - 8a 2 6 2 



16a^(l + 6)" (3) 



(1 _)-2( a _^) 



64a*(-a + a 2 + 3 ) ,, 



" (1 - a 2 ) 5 (1 + 4a - a 2 ) 



1' 2 i~\ I 9\9 / -i t f~\ y 9 *-*" *l i 4A9 



_i_ \ \ (1 + a ) (1 4- 2a 6a 2 2a d + a*r / K \ 



X / /I a 2 ^ 5 n -I- 4a a 2 ) ' '' 



Now, as in the < Archiv,' III., 1, p. 75, with the 6 there replaced by - l + a and 



1 a? 



I - ' 



