425 
1909-10.] Tlie Theory of Persy mmetric Determinants. 
equations in the %’s together with the equation from which the set was 
derived, and eliminates the B’s, the result in the case of r = 3 being 
1 
5 o 
s i 
S 2 
whence of course he deduces : 
1 x 
and the alternative form 
n 2 
a i 2 a 2 
n ( n - 1 )a 1 
x? -D 2 
'2 S 3 
>3 *‘4 ^-3,1 
»r D. 2 the expression 
X 2 
/AM 
1 
X 
h 
or ( — i ) 
\ v 
s o 
S 1 
S 2 
S 3 
s i 
^2 
h 
<q 2a 2 3 a 3 
2 • n (n l)oq 2)^ 
1 x + cq x 2 + a x x 4 - a 2 • 
The process of finding f r is quite similar to this but much more trouble- 
some, the equation taken along with the set of equations in the s’s pre- 
paratory for elimination being 
where 
/r(«0 
/ 0 ) 
B r _-^ r U 0 br— 1 , r— 1 • • • • "4* B? — 1 
+ 
X-. x-x 0 
+ 
+ 
x - x r , 
The two previous steps necessary to reach this are 
/r(s) = fr{Xi) = ^ D,-i(av) 
i=l i=l 
and the result of the elimination is 
s o 
*1 
■ . S r _! 
s i 
s 2 
. . S r 
/A*) = 
K 
A X ) 
A r 
S r- 2 
s r-i . , 
■ • S 2r-Z 
^0 
«q 
• . u r _ x 
which in the case of r = 4 is changed by means of the substitutions 
u m = XU m - 1 ~ S m _i , u Q = f\ { x y~ r f( X )i 
