^{page 181 note *} See especially p. 288.

^{page 188 note *} The paper, as it appears in Crelle's Journal, is disfigured by misprints, which have not been fully corrected in the Collected Math. Papers.

^{page 189 note *} Apparently it is meant to be implied that each of the numbers occurs only once in the expression.

^{page 190 note *} Supplying this defect we see that in strict accordance with Cayley's definition

—a function of twelve variables which is not a determinant in the acceptation either of the present time or of the time preceding Cayley.

^{page 193 note *} And of course without loss of generality, as Cayley might have said.

^{page 195 note *} There is herein used the fact, first noted by Rothe in 1800, that the cofaotor of rs in any determinant is equal to the cofactor of sr in the conjugate determinant.

^{page 196 note †} Along with this fact Spottiswoode associates the statements that u ₁ + u ₂ + … + u _n = 0, v ₁ + v ₂ + … + v _n = 0, which are manifestly incorrect.

^{page 199 note *} All of them fall to be dealt with when giving the history of the development of the theory of determinants in general.

^{page 201 note *} A serious misprint in the original is here corrected.

^{page 207 note *} Since the left member is what Cayley called a “bordered skew symmetric determinant”; and since, as Jacobi noted, a differential-quotient of H with respect to one of its elements is a function of the same kind as H, we have here one half of Cayley's proposition that a bordered skew symmetric determinant is expressible as the product of two Pfaffians.

^{page 211 note *} It ought to be noticed also that Baltzer uses the equation

to verify Spottiswoode's theorem for the case where Δ is odd-ordered, the reasoning being that as Δ is then kaown to be zero, so also must ∂Δ/∂_ars, and that therefore A_rs = A_sr.