392 Proceedings of Royal Society of Edinburgh. 
and as illustrating (a), (c) and (e) we have 
[sess. 
+ CO 
a 2 + to 
a 3 + co 
a 4 + co 
b 4 + co 
b 2 + co 
b 3 -f- co 
b 4 + co 
+ co 
c 2 + co 
Cg + CO 
C 4 + co 
d 4 + co 
d 0 + co 
d 3 + co 
d 4 + co 
| af 2 c 3 d 4 | + w| af 2 c 3 1 | + o)| a 1 b 2 ld 4 | + 
I af> 2 c 3 d 4 | — 
1 1 
. 1 
1 a , 
1 \ \ h h 4 
1 c 1 c 2 c 3 c 4 
1 d 1 d 2 d 3 d 4 
(5) Returning again to the result (VIII), and multiplying both 
sides by dx , we obtain with the help of (A) § 1 
1 dx{\ — acx 2 ) 
1 - bx + acx 2 + (1 + ax)( 1 + cx)( 1 - bx + acx 2 ) 2 
= 1 + P X X + P 2 X 2 + /? 3 « 3 + . . . . 
- d 
H-J 
IT 
l 
. 1 1 
x 2 - d 
.111 
1 b | 
1 b a 
1 b a . 
1 c b 
1 c b a 
1 . c b | 
x 3 - . 
and, therefore, by theorem (e) of § 4 
= 1 + (b + d)x + 
b + d 
a+d 
x 2 + 
b + d 
a + d 
d 
c + d 
b + d 
c + d 
b + d 
a + d 
d 
c + d 
b + d 
x 6 + 
(XV) 
From the same two sources we similarly have 
/ x{l - acx 1 ) 
1 -bx + acx 2 (1 + ax)( 1 + cx)( 1 -bx + acx 2 ) 2 
x 2 + 
I / 1 
e + 
/ 
1 
1 
1 si 
1 
b 
a 
1 
c 
b 
(XVI) 
This is due to Dr F. S. Macaulay, v. Math. Gazette , ii. p. 61. 
( Issued separately February 28, 1903.) 
