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XVIII.—The Less Common Special Forms of Determinants up to 1860
Published online by Cambridge University Press: 15 September 2014
Extract
There now only remain for consideration those special forms which, prior to 1860, had not received any noteworthy attention. These will be found to include: (α) permanents, which are touched on by three authors; (β) determinants with the typical element ars + brsi, which are referred to in four memoirs; (γ) two other forms, which are each dealt with in two papers; and (δ) nine others, which make their appearance only once.
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- Copyright © Royal Society of Edinburgh 1912
References
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 +.