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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
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(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
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
