1. If all the negative terms of the determinant ∣ a1b2c3 … ∣ be changed in sign, we obtain a symmetric function, dealt with by Borchardt and Cayley, known as a Permanent and denoted by

The more important elementary properties of such functions are given in a paper published in the Proc. Roy. Soc. Edin., xi. pp. 409–418. As might be expected, relations are found to exist between them and determinants, an important instance being the theorem of § 7 of the said paper. Another theorem, not hitherto noted, deserves now to be put on record.

2. For the case of the 2nd order it is

the truth of it being self-evident.

For the case of the 3rd order it is

which is easily verified by observing that the coefficients of a1, a2, a3 in the expression on the left-hand side are respectively

and that by the previous case each of these vanishes.