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X.—The Relations between the Coaxial Minors of a Determinant of the Fourth Order

Published online by Cambridge University Press:  06 July 2012

Thomas Muir
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Extract

1. The existence of relations between the coaxial minors of a determinant was first discovered by MacMahon in 1893. The whole literature of the subject is comprised in three papers, viz.:—

MacMahon, Phil. Trans., clxxxv. pp. 111–160.

Muir, Phil. Mag., 5th series, xli. pp. 537–541.

Nanson, Phil. Mag., 5th series, xliv. pp. 362–367.

My present object is to continue the investigation of the relations in question, and more particularly to draw attention to an explicit expression for a determinant of the 4th order in terms of its own coaxial minors. At the outset some fresh considerations regarding determinants in general will be found useful.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 39 , Issue 2 , 1900 , pp. 323 - 339
Copyright
Copyright © Royal Society of Edinburgh 1900

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References

page 327 note * In the case of each expression under a root-sign a certain amount of simplification is also possible, e.g., we find

page 331 note * Instead of the biquadratic of this section another might readily have been obtained from the single equation numbered (9) in Professor Nanson's paper, viz.,

where

μ = A+B+C+D-½Δ−1−BC−CA−AB.

Observe also that this equation gives a much simpler expression for Δ, viz.:—

page 334 note * In effect the substitutions are the same as the circular substitution if we consider cos(β+γ) cos(γ+α), cos(α+β) as invariant.

page 335 note * It is interesting to note the mode in which the more general relation connecting cos (α+β+γ), cos α, cos β, cos γ, passes over into this on putting γ = 0 in the former. The result of the substitution is

where the elements of the 4th column are easily transformed into zeros with the exception of the last element which becomes

1 + 2 cos α cos β cos (α + β) − cos2 (α + β) cos2 β − cos2 α,

so that the value of the determinant is seen to be

With this mode of degeneration may be compared that seen on p. 377 of Proc. Roy. Soc. Edin., xx

page 336 note * See Quart. Journ. of Math, xviii. pp. 170, 171.

page 339 note * Proc. Roy. Soc. Edin., xxi. p. 333.