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XXIX.—On the Applications of Quaternions in the Theory of Differential Equations

Published online by Cambridge University Press:  06 July 2012

J. H. Maclagan-Wedderburn
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Extract

The object of this paper is twofold: in the first place, to classify and sytematise vector differential equations; and in the second place, to show the applicability of quaternions to the theory of differential equations. So far as I am aware, the only paper of importance on the subject is one by Tait. He, however, deals only with simple cases.

In the classification of forms I have followed in general the treatment adopted by Forsyth. Reference will also occasionally be made to Jordan. In what follows the order of the highest occurring differential coefficient will be called the rank of the equation; while the order of the equation will be used, as usual, to denote the order of the equivalent normal system.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 40 , Issue 4 , 1905 , pp. 709 - 721
Copyright
Copyright © Royal Society of Edinburgh 1905

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References

page 709 note * “Note on Linear Differential Equations,” Proc. R.S.E., 1870. Scientific Papers, vol. i. p. 153Google Scholar.

page 709 note † Treatise on Differential Equations, 1888.

page 709 note ‡ Cours d'Analyse, i.–iii., 1887.

page 710 note * Proc. R.S.E., xxiv., 1903, p. 410Google Scholar.

page 713 note * Note on the form of eϕ.—If the elements of ϕ are we have

and in general

f can also be expanded in the form Aϕ2 + Bϕ + C where A B and C are given by the equations

—(See Tait, Quaternions, 3rd ed., p. 124.

page 718 note * See Prof. Chrystal, “A Fuudramental Theorem regarding the Equivalence of Systems of Ordinary Linear Differential Equations,” Trans. R.S.E., 1895.