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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.:—
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU1.gif?pub-status=live)
The mode of establishing these may be illustrated by proving the last of the three.
By a well-known therem we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU2.gif?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU3.gif?pub-status=live)
and therefore by cyclical substitution
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU4.gif?pub-status=live)
From these by addition there results
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU5.gif?pub-status=live)
The three fundamental identities which can be treated in this manner are
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU6.gif?pub-status=live)
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:—
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU7.gif?pub-status=live)
and this expanded form may, by the use of the theorem
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU8.gif?pub-status=live)
be changed into
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU9.gif?pub-status=live)
and thus into
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU10.gif?pub-status=live)
so that by a second use of the said theorem we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU11.gif?pub-status=live)
and finally
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU12.gif?pub-status=live)
or by a third use of the same theorem
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151109113621776-0160:S0080456800034207_eqnU13.gif?pub-status=live)