208 Proceedings of Royal Society of Edinburgh. [sess. 
and since, as we have seen, it is permissible to substitute 
P 02p f or / 9P V 
ba r fa ss \da rs J 
there results 
( 5 2 P Y_ + 0H . 
V3« rr 3a g / “ da rs ; 
so that the expansion for P above obtained may be altered into 
from which by extraction of the square root we have 
- 2.(0 
This will be recognised as a third mode of writing an already 
well-known result, and, as Brioschi notes, gives a property of 
the function H similar to a property of determinants (“ la quale 
equazione contiene una propriety della funzione H analoga ad una 
nota dei determinant! ”). 
Prom this he passes to what he calls the characteristic property 
of H, viz., its change of sign consequent upon the transposition of 
two indices. Calling H' what H becomes when r and s are inter- 
changed, he notes that in those terms of H in which the element 
a rs occurs there can be no other element with the same indices, and 
that therefore 
0H J f_0H' 
da rs da rs 
Then since the same interchange made in P leaves P in reality 
unaltered, — that is to say, since H 2 = H' 2 , — he obtains 
h 3H = h ,?H\ 
bct rs da rs 
and, it having been shown that the two differential-quotients here 
appearing are of opposite signs, it follows that so also are H 
and H\ 
Lastly, he passes on to skew determinants in general ; and, using 
