^{page 313 note *} By an oversight three terms of this are left out by Joachimsthal.

^{page 316 note *} These may be established as follows. By separating the terms of K which involve a from those which do not, we see that

a determinant differing from Φ in the first row only, and consequently on multiplying by a and adding we obtain

^{page 317 note *} This second form of Φ may be got directly from the determinant by expanding in terms of the two-line minors formable from the first and third columns, and the minors complementary to these. of course we also have

^{page 322 note *} As (01) occurs only in the element σ₁ − (11), its cofactor is the primary minor obtained by deleting the first row and the first column, and this is seen to be by definition.

^{page 323 note *} Probably the easiest way is to express the determinant as a sum of determinants with monomial elements. In the case of the third order the number of such determinants is 64, of which 40 vanish, the sum remaining being

where rst stands for the determinant whose columns are in order, the r ^th, s ^th, t ^th columns of the array

^{page 326 note *} The minus sign is omitted by him throughout. If the number of x's had been odd, the sign would have been +.