1902 - 3 .] Dr Muir on Generating Functions of Determinants. 389 
This determinant, however, clearly contains the factor a + 6 + c, 
or a 
say 
being 
1 
a 
• . . . 
... c 
1 
b 
a 
1 
c 
b 
a 
. . 
1 
c 
b . . . 
1 
, 
b a 
1 
. 
c b 
r 
<T 
<T 
<x . . . 
<T 
1 
b 
a 
1 
C 
b 
a . . . 
• 
1 
c 
b ... 
1 
. . b 
a 
1 
c 
b 
. 
1 
1 
1 . 
. . . . 1 
1 
1 
b 
a 
. 
1 
c 
b 
a 
1 
c 
b . 

• 
1 
.... b 
a 
1 
• 
. . . . c 
b 
(V) 
if the determinant resulting from bordering on the top and 
left-hand sides with one zero and r - 1 units he denoted by 
B r _ 1 . Substituting the successive values of this corresponding 
to r= 1, 2, 3, .... in place of the coefficients of x°, x 1 , x 2 , ... . 
above we have 
a 
1 + ax 1 
c 
h 
4 - cx 
b - 2 acx 
1 - bx + acx 2 
< t + cr(2^ 1 + + or(3/? 2 + crB 2 )a? 2 -f . , . . 
But the left-hand member 
Consequently there results 
1 — acx 2 
(a + 6 + c)( 1 - acx 2 ) 
(1 + ax){ 1 + cx)( 1 - bx + acx 2 ) 
— X/1 xn j -o\ — l + (2j3 1 + (T^ 1 )x + (3/3o + a-B 9 )x 2 + ... (YI) 
(1 +aa;)(l +cx)(l - bx + acx 2 ) v ri v v r2 v v f 
By a different treatment of the unsymmetrical determinant 
which is the cofactor of o- in (IY) an interesting alternative 
form of expression for the coefficients of this expansion is 
obtainable, viz. : 
- (a + c)/J r _ 2 + (a? + <*)&_ 3 - (a 3 + c 3 ) / 8 r _ s + . . . ( VII ) 
