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III.—On the Eliminant of a Set of General Ternary Quadrics.—(Part II.)

Published online by Cambridge University Press:  06 July 2012

Thomas Muir
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Extract

(26) Of the various determinant forms thus far obtained the most promising is that of §8 or that of §14; and to these it is desirable now to return in order to obtain an expression for the eliminant in the ordinary non-determinant notation. In doing so it will also be well to make a slight change in the coefficients of the three quadrics, viz., to write f, g, h for 2f, 2g, 2h, as in this way the diversity in the cofactors of the determinants occurring in the last three rows of either form of the eliminant disappears.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 40 , Issue 1 , 1905 , pp. 23 - 38
Copyright
Copyright © Royal Society of Edinburgh 1905

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References

page 27 note * For each of the terms an alternative form is available, by reason of the existence of curious kind of identity of which there are three instances, viz.:—

The mode of establishing these may be illustrated by proving the last of the three.

By a well-known therem we have

where, be it observed, each side consists of two terms of a traid. Multiplying, then, both sides by the remaining term of either traid, say by 84ʹ, we have

and therefore by cyclical substitution

From these by addition there results

The three fundamental identities which can be treated in this manner are

or, of course, their derivatives by cyclical substitution.

page 29 note * Muir, T., “Further Note on a Problem of Sylvester's in Elimination,” Proc. Roy. Soc. Edin., xx. pp. 371–382Google Scholar.

page 32 note * The result obtained by Lord M'Laren, in his paper on “Symmetrical Solution of the Ellipse-Glissette Elimination Problem,” in the Proc. Roy. Soc. Edin., xxii. pp. 379—387Google Scholar, is the particular case of this where f 1, f 3, g 2, g 3 are made to vanish and a 1, a 2, a 3, are put equal to b 2, b 1, b 3 respectively.

page 35 note * See Muir's “Determinants,” p. 216, ex. 7. A more general theorem is obtained thus:—

and this expanded form may, by the use of the theorem

be changed into

and thus into

so that by a second use of the said theorem we have

and finally

or by a third use of the same theorem