1902-3.] Dr Muir on Generating Functions of Determinants. 387 
The Generating Functions of Certain Special 
Determinants. By Thomas Muir, LL.D. 
(MS. received September 8, 1902. Read November 3, 1902.) 
(1) From a general theorem, known since 1855,* and perhaps 
earlier, regarding the reciprocal of the series 
1 + a x x + a 2 x 2 + a s x s 4 - 
it follows that 
1-fe + OCT 2 = Po + ft* + ft* 2 + ft* 3 + • • • ( A ) 
where /3 0 = 1 and 
Pn = 
b 1 
= 
b a 
ac b 1 
c b a, 
. ac b 1 
. c b a 
. ac b 
. . c b .... 
This at once leads to 
bx - acx 2 „ ^ ^ „ 
l-bx + ae* = ft* + £ 2 * + ft* + > 
(b 2 - ac)x 2 - abcx 3 
1 -bx + a^~ = ^ + ft* 3 + ' 
and generally, by subtracting 1 + p 1 x-\-p 2 x 2 + . . . + to 
- ac/? r _ 1 3r +1 _ „ o 
1 — bx + (zcic 2 
•r+Y 
*r + 1 
+ 
or 
- acB r _,x 
1 - bx + acx 2 = + &+1 35 + &+2^ 2 + • • 
* Fatjre, .... Theoreme sur la somme des puissances semblables des 
racines. Nouv. Annales de Math., xiv. pp. 94-97. 
