180 
Transactions of the Royal Society of South Africa. 
produces 
(a, + ft)5 ... (ax 4-/35)5; 
(a5+/3x)5 ... (a5+/3^ 
(HI.) 
6. Taking these results along with Scott's of 1879 {Messenger of Math. 
viii. pp. 182-187), we obtain a remarkable identity — possibly the first 
observed of its kind — giving an expression for a determinant in terms of 
a permanent, that is to say, a function of one class in terms of another of 
the directly opposite class. Thus, for the fourth order, we have 
2/3x 
a, + p, . 
-4 
= 4 
. . ttz + 
Saittz . — 6 
a. + jJi . 
.. O3+/34 
-4 . 
«4+/3x . 
.. «4+/34 
the connecting factor, which is here 4, being for the nth order 
/_1Yi"(n-.) fhPh^^K 
^ ' ' 1.2... 7r 
where n^^n{n — l) ... (7^ — 7' + 1)/1.2 ... 7'. 
(IV.) 
7. A direct mode of establishing this identity is something to be 
desired. All that we can suggest as a substitute is a proof that the two 
members of it have the same final development. Taking, for example, 
the permanent of the third order 
+ + 
«2 + /3i a2 + /32 a2 + /33 
a3 + /3, a^ + ft^ a3+/33 
and recalling the fact that the law for the partitionment of determinants 
with polynomial elements holds also for permanents we obtain 
+ + + + + + 
«! fti ai 
«i «i ft 2, 
I /32 /33 
«2 «2 «2 
+ rtg a2 /33 
+ ... 
...+ \ft 
X ft. ft. 
03 03 
1^3 "3 ft 3 
\ft 
X ft. ft. 
