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XVII.—Quaternion Investigation of the Commutative Law for Homogeneous Strains
Published online by Cambridge University Press: 15 September 2014
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1. If a plastic body be subjected to a uniform change of shape, any two equal, similarly placed cubes become equal, similarly placed parallelepipeds. It is well known that the character of the deformation may be determined by a set of nine constants; but the physicist naturally prefers to regard the strain as an operator, and to represent it by a single symbol. The resulting operational algebra may even react favourably on the mathematical aspect of the question, and give us new methods of attack for old problems.
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- Copyright © Royal Society of Edinburgh 1915
References
page 170 note * A general law for commutation of matrices is given by Taber, H., Am. Acad. Proc, 1891, 26, 64–66CrossRefGoogle Scholar, but the result is ill adapted to physical interpretation. Several elegant examples are given by Gibbs, in the language of dyadics (Scientific Papers, ii, p. 63), but the discussion is not completed so as to cover all cases.
C. J. Joly, in his Manual of Quaternions, gives the rule: Two linear vector functions are commutative when, and only when, they have the same axes. This rule holds whenever both strains have three distinct axes, but is otherwise insufficient, as exemplified in the text.
page 170 note † Kelland, and Tait, , Introduction to Quaternions, chap. x.Google Scholar
page 171 note * Kelland and Tait, loc. cit.
page 171 note † Tait, , Quaternions, 3rd ed., art. 176.Google Scholar
page 171 note ‡ Lectures on Quaternions, (7), p. 567.
page 173 note * Tait, , Quaternions, 3rd ed., p. 299.Google Scholar The axes in Tait's example are the axes of the pure part of the strain.
page 179 note * Phil. Mag., June 1902, p. 579