250 Proceedings of the Royal Society of Edinburgh. [Sess. 
studied, and indeed was completely solved by Heilermann* so long ago as 
1845. His result is that the coefficients b, a v b v a 2 , b 2 , . . . in the continued- 
fraction are simply expressible in terms of determinants of the kind called 
by Sylvester per symmetric, namely, 
1 b a) | 
1 b {1) b {2) 
b il < b m 
5d) 5(2) 5(3) 
V' V s ’ 
5d) 5(2) 5(3) 
, 
b m 
5 
5(2) 5(3) 5(4) 
5(2) 5(3) 5(4) 
5(3) 5(4) 5(5) 
which are formed of the coefficients in the given power-series. 
On comparing Heilermann’s result with that of § 2, it is evident that 
a connection must exist between persymmetric determinants and the 
coefficients in powers of matrices of the form (4). This connection is 
easily shown by aid of Stieltjes’ theorem. For if we denote the quadratic 
form (11) by A, we see that the persymmetric determinant 
5 (1) 
5(2) 
5(2) 
5(3) 
5(3) 
5(4) 
is 4 X the Jacobian of 
0A 
ax. 
0A 
ax. 
ax 
ax n 
with respect to X 0 , X 15 X 2 . But 
by (12) and (14) A may be written in the form 
A = P 2 + cqQ 2 + + . . . 
where 
= a 00^0 -t a 01^1 -t a 02^2 “t • • ' 
Q == a 11 X 1 + a 12 X 2 "t a i3^3 + • • • 
R = a 22 X 2 + a 23 X 3 + . . . 
and as Q does not contain X 0 , R does not contain X 0 or X t , etc., we see 
that the Jacobian has the value 
0 2 A 
0 2 A 
0 2 A 
/ 0P N 
y 0R v 
0P 2 
0Q 2 
0R 2 
\0X O , 
' \sx,/ 
\0X 2 / 
Thus we have the result that if b (1) , b (2) , b (3) , b (4) are the leading coefficients 
in the first, second, third, and fourth powers of the matrix 
* Journal fur Math., 33 (1845), p. 174. 
