142 Proceedings of the Royal Society of Edinburgh. [Sess. 
inequalities. Our present interest in the demonstration, however, lies 
merely in the fact that it is based on two properties of axisymmetric 
determinants which it is desirable to isolate and to have more carefully 
formulated than it was Sylvester’s wont to do. They are — 
(1) If | (11) (22) .... ( nn)\ be axisymmetric, and the result of 
multiplying it by itself be | [11] [22] .... [jm]|; and if f(x) be the 
determinant got from the former by adding x to each element of the prin- 
cipal diagonal, and F(x) the determinant got similarly from the latter ; then 
/(*)•/(-*) = F(-*2). 
(2) If F(aj) be expanded and arranged according to descending powers 
of x, so that 
F(jb) = x n + Ci # 71-1 + C 2 # n ~ 2 + ... + C n , 
then C r is the sum of the squares of all the r-line minors of the original 
determinant, it being understood that the one-line minors are the elements 
and the Ti-line minor the determinant itself. 
Both are taken for granted, — a liberty which is not so defensible in the 
second case as in the first ; for C r is at the outset obtained merely as the 
sum of the r-line coaxial minors of | [11] [22] .... \nri\ |, and use must 
thus be latently made of the not quite self-evident theorem that if A be an 
axisymmetric determinant, the sum of the r-line coaxial minors of A 2 
is the sum of the squares of all the r-line minors of A. As an illustration 
of the whole, let us take the case where the given determinant and its 
square are 
a 
h 
9 
L 
R 
Q 
h 
b 
f 
and 
R 
M 
p 
9 
f 
c 
Q 
P 
N 
and where therefore 
We have then 
I d 2 + h 2 + g 2 all + hb +fg ag + lif + gc 
h 2 + b 2 + f 2 hg -t-bf+fc 
gt+ft + c 2 . 
a + x 
h g 
a- x 
h 
9 
L-x 2 
R 
Q 
h 
b + x f 
• 
h 
b-x 
' f 
= 
R 
M - x 2 
P 
9 
f c + X 
9 
f 
c — X 
Q 
P 
N -X 2 
+ a^L + M + ff) - x 2 ( 
,/\LB 
LQ 
M P|\ 
VIr m| + 
Q N 
+ 
p nI/ + 
L R Q 
R M P 
Q P N 
