of Edinburgh, Session 1881-82. 
417 
simple deduction which can be made from it. The number of de- 
terminants on the right is then only two. Taking the reciprocals 
of a , b , c, . . . for the elements of the factors on the left we have 
+ 
1 
i 
1 
1 
i 
1 
1 
i 
1 
1 
1 
1 
a 
b 
c 
a 
b 
c 
a? 
b 2 
c 2 
ag 
bh 
ck 
1 
l 
1 
1 
1 
1 
1 
l 
1 
+ 2 
1 
1 
1 
d 
e 
/ 
d 
e 
/ 
d 2 
e 2 
/ 2 
da 
eb 
fc 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
g 
h 
k 
g 
h 
k 
g 2 
7i 2 
k 2 
gd 
he 
¥ 
The last determinant here, however, 
cl e f 
g h k 
a b c 
So that if any determinant of the tim'd order with non-zero elements 
vanishes identically , the product of the Permanent and Determinant 
whose elements are the reciprocals of the elements of the said determi- 
nant is equal to the determinant whose elements a, re the squares of 
these reciprocals. For example, we have identically 
= ( abcdefghk ) _1 
a 
a + d 
a -P 2 d 
a + 3d 
a + id 
a + 5d 
a -P Qd 
a + 7d 
a + 8d 
= 0 ; 
hence 
+ 
,-i 
a~* (( a + d)~ x (a-p2cZ) _1 
( a + 2>d)~ 1 (a -l- id) -1 (n-p5c?) -1 
(a + Qd)' 1 (a + 7d)~ 1 (a-pS^) -1 
+ 
> — i 
a~ x ( a + d ) _1 ( a + 2d )~ 1 
( a + 2td)~ l (a-p4c7) _1 ( a + 5d ) -1 
(a + 6d)~ 1 (a+7 d)' 1 (a + 8d)~ l 
a~ 2 ( a + d )" 2 (a + 2(i)“ 2 
(a + 3d)~ 2 ( a + id )~ 2 (a + 5d) 
(< a + 6d )- 2 (■ a + 7d )~ 2 (a + 8d) 
-2 
11. As the interchange of rows and columns has no effect either 
in Permanents or Determinants, it is evident that we may have 
more than one form for the developments in §§ 5, 6, 7. From the 
