84 
Proceedings of the Royal Society of Edinburgh. [Sess. 
1 a \ C 2 ! 
! a \ C 3 1 1 a i C 4 1 
1 «2 H 1 ! a i C 4 1 1 “3 C 4 1 
|/i 9 A 
l/l ^ 2 ] l/l G 
1 9\ ^ 2 1 1 9\ ^ 2 1 1 ^ 2 1 
1 m 2 
«sl 
\AK\ 
1 m 2 
A 1 
l/l 94, 1 
IA'uM- wm 
\KK\ 
! m 2 
0'3 1 
1 A 94 1 
I /3 ^4 
\ } h\\ 
1 r 2 
S 3 1 . 
where the 6 + 36 + 6 two-line determinants are those formable from the 
three arrays of the expression with which we started. 
Similarly, if the given determinant be the product of four determinants 
A x , A 2 , A 3 , A 4 , it is seen that the two-line minor belonging, say, to the 
2nd and 4th rows and to the 1st and 2nd columns of the product is the 
bipartite function * representable briefly by 
Two-line minors formable 
from rows 2 and 4 of A 1 
Two-line minors formable 
from columns 1 and 2 of A 4 
Two-line minors formable 
from conjugate of A 2 
Two-line minors formable 
from A 3 
and so on generally. 
6. In at least one case the two-line minor of a product-determinant is 
of very special interest. This is when the second determinant of the 
product A 1 A 2 A 3 is the adjugate of another determinant, the reason being 
that important changes are then possible on both sides of the identity. 
Thus, taking the product 
I a l ^2 ^3 ^4 I ' i -^1 ^*2 -^-3 ^4 I I I m \ n 2 r S S 4 I 
we find, in the first place, that each element of the product-determinant is 
itself expressible as a determinant; for example, for the element in the 
place 1,1 
a i 
a 2 
«3 
«4 
H, 
m Y 
F 2 
g 2 
H 2 
n i 
F 3 
g 3 
h 3 
k 3 
r i 
g 4 
h 4 
k 4 
s i 
* In the original memoir (Trans. Roy. Soc. Edin., xxxii. pp. 461-482), in which the 
properties of the functions which represent the elements of product-determinants are in- 
vestigated, it is stated (p. 481) that they were given the name ‘ bipartite ’ from Cayley’s use 
of the word for a special set of the functions, namely, those of the third degree. It has often 
since appeared to me that it would have been preferable to have extended the meaning of 
another word, namely, the word £ cumulantj this being the name given by Sylvester to a 
special set of the second order. 
