390 Proceedings of Royal Society of Edinburgh. 
Thus, to take the case of the 5 th order, 
[sess. 
1 
a 
c 
. 
a 
c 
1 
b 
a 
l 
b 
a 
1 
c 
b 
a 
— A + 
l 
c 
b 
a 
1 
. 
c 
b 
a 
l 
c 
b 
a 
1 
c 
b 
l 
c 
b 
= p 4 -(a + c)p 8 - 
. cl 1 
1 b 
1 c 
1 . 
= ^4 “ ( a + C )/ 5 3 + ( ft2 + + 
a 
b a 
c b 
c 
1 l 
1 i 
= & - (a + c)/? 3 + (a 2 + c 2 )/3 2 - (a 3 + c 3 )/^ + (a 4 + c 4 
(3) The right-hand member of (YI) may evidently be written 
as the sum of 
1 + 2 + 3 j3. 2 x 2 + 4^ 3 ^ 3 + 
and o^B-,# 4- B 2 a? 2 + B 3 £ 3 + ....), 
the former of which is known from (I) to be equal to (1 - acx 2 )/ 
(1 - bx + acx 2 ) 2 . We are thus able to conclude that 
1 - acx 2 
(1 + ax){l +ca?)(l - bx + acx 2 ) 2 ~ 1 2 ^ &X ~ 
1 • l ! - 
. 1 1 
x — 
.111 
1 h 
1 b a 
e 
T— H 
1 c b 
1 c b a 
1 . c b 
(4) Each of the peculiar determinants in (VIII) is the negative 
sum of the signed primary minors of the particular /3 from which 
it is constructed by bordering. At this point, therefore, the 
following propositions regarding the sum of the signed primary 
minors of a determinant are of interest : some of them, indeed, 
are immediately useful. 
(a) An expression for the negative sum of the signed primary 
minors of any determinant is got by taking a determinant of the 
next higher order , whose first element is zero , ivith the given 
