The Algebra of Invariants
The Algebra of Invariants
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In our eReader you can find the full English version of the book. Read The Algebra of Invariants Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book The Algebra of Invariants
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Thus if PQ pass through T, and RS be the polar of T, the quadratics corresponding to PQ, RS are harmonic, or in other words, when two quadratics are harmonic, the corresponding lines are conjugate with respect to the conic.
This can be easily verified by using the tangential equation of the conic.
Consider a binary quartic representing four points A, B, C, D on the conic, and let BC, AD meet in E, CA, BD in F, and AB, CD in G.
Then the quadratic corresponding to the polar of E is harmonic to th...e two quadratics corresponding to AD and BC, i. E. This polar meets the conic in two points which are the double points of the involution having B, C and A, D for conjugate elements.
Thus the polars of E, F, G meet the conic in the double points of the three involutions determined by the four points A, B, C, D; but EFG being a self-conjugate triangle of the conic, the polar of E is FG, and the lines EF, FG for example are conjugate lines with respect to the triangle, hence the pairs of double points of any two of the three involutions are harmonically conjugate.
What to read after The Algebra of Invariants?
You can find similar books in the "Read Also" column, or choose other free books by J H John Hilton Grace to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Thus if PQ pass through T, and RS be the polar of T, the quadratics corresponding to PQ, RS are harmonic, or in other words, when two quadratics are harmonic, the corresponding lines are conjugate with respect to the conic.
This can be easily verified by using the tangential equation of the conic.
Consider a binary quartic representing four points A, B, C, D on the conic, and let BC, AD meet in E, CA, BD in F, and AB, CD in G.
Then the quadratic corresponding to the polar of E is harmonic to th...e two quadratics corresponding to AD and BC, i. E. This polar meets the conic in two points which are the double points of the involution having B, C and A, D for conjugate elements.
Thus the polars of E, F, G meet the conic in the double points of the three involutions determined by the four points A, B, C, D; but EFG being a self-conjugate triangle of the conic, the polar of E is FG, and the lines EF, FG for example are conjugate lines with respect to the triangle, hence the pairs of double points of any two of the three involutions are harmonically conjugate.
What to read after The Algebra of Invariants?
You can find similar books in the "Read Also" column, or choose other free books by J H John Hilton Grace to read onlineMoreLess
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