409 
1909-10.] The Theory of Persymmetric Determinants. 
1 
X 
z 2 
. . x m 
v l 
• • 
% 
v 2 
v z 
• • ^m+1 
v m - 1 
V m 
v m+ 1 • 1 
• * V 2171-1 
or, by further putting w = v p+l — xv p , 
W 0 w 1 ... w m _ l 
W 2 ... w m 
W m —\ W m . . . W 2m _ o . 
After finding other forms for M(&), and varying (§ 2) the mode of 
finding them, Jacobi proceeds (§ 3, pp. 140-146) to deal with N(as), first 
remarking, of course, that the one function is immediately determinable 
from the other, because the problem of representing u x , u 2> ... . by 
N(cc)/M(zp) is the same as the problem of representing , u 2 l , .... by 
M(aj)/N(£c) . Instead of utilising this, however, he takes from the theory 
of “ partial fractions ” the result 
i=n+m + 1 
N(«!) = V Nfe) 
f( x ) ^s^ Xi ~ *)/'(*<) ’ 
•whence follows 
i=n+m + 1 
~8(x) = V uMxJ 
/(*) ~~ / ' fa. - »)/'(*«) ’ 
«=1 
and therefore from the data that 
1 
X 1 
X. I 2 
+ a x x 4 + a 2 x 2 ) 
1 
x 2 
x 2 2 
UtpCyp^ tX() "f* CL-pC 2 QLyX 2 ') 
1 
x 3 
x 3 2 
u 3 x 3 P(a 0 + ai x 3 + a 2 x 3 2 ) 
1 
x 4 
X 2 
u 4 X 4 P(a 0 + a x x 4 + a 2 X 2 ) 
- 0 when p = 0 or 1 , 
or, what is the same thing, 
1 
x i 
x 4 2 
U^P 
1 
Xi 
X\ 
U l X 1 P+ 1 
1 
X 1 
Xj 2 
U- [ X l P+‘ i 
1 
x 2 
x 2 
u 2 x 2 p 
1 
x 2 
X 2 
U 2 X 2 P +1 
a l + 
1 
x 2 
x 2 2 
U 2 X 2 P+ 2 
1 
x 3 
/y» 2 
U 3 X 3 P 
a 0 + 
1 
x 3 
*3 2 
u 3 x 3 P+ 1 
1 
x 3 
x 2 
u 3 x 3 p+ 2 
1 
x 4 
X 2 
U{X 4 P 
1 
x 4 
X 2 
U 4 X 4 P+ X 
1 
x 4 
x 4 2 
U 4 X 4 P + 2 
0. 
From these two equations on solving for a 0 : a x : a 2 and substituting in a 0 + a x x + a<p? we 
obtain 
1 
X 
X 2 
M(») = 
v o 
”1 
"2 
1 i/j 
v 2 
u 3 
where vP = | x-^ x 2 x 3 2 x 4 Pu 4 | , or 
M(£C) = 1 
w 0 
"1 
1 
COo 
where oo p = v p +i - xv p = I x 4 ° xM x 3 2 x£u 4 (x 4 - 
x) 1 
