Plane Cubics With a Given Quadrangle of Inflexions
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The equianharmonic properties, however, are simply a consequence of harmonic properties; for if we write x = 0, y = 0; x + Py = 0, x — ay = 0; x — (3y = 0, x + ay — 0, * Consider x = 0, x * >/3y = 0, x + iy = 0. Let X = x + V%, Y = x — V3?/; then x = 0, x + iy = are transformed respectively into X + Y = 0, X — «F = 0; and the cross-ratio of the four lines X = 0, 7 = 0, Z + Y = 0, Z — uY = is — w 2, where w 3 = 1. Similarly for the other combinations.
266 Turner : Plane Cubics ivith Inflexions.
...and impose the condition that x = 0, x + (3y = be harmonic with respect to x — fiy = 0, x + ay = 0, we find /3 = 3a ; and the condition that x + 3a?/ = 0, x — ay = be harmonic with respect to x ± iy = shows that 3a 2 = 1.
Similar statements hold for the lines through B; and also for the set through A, except that in this case the involution is hyperbolic. Con- sequently the sides of the three quadrangles, and hence the vertices, are uniquely determined from the sides of the base quadrangle and its diagonal triangle by means of harmonic properties.
What to read after Plane Cubics With a Given Quadrangle of Inflexions?
You can find similar books in the "Read Also" column, or choose other free books by Bird Margaret Turner to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The equianharmonic properties, however, are simply a consequence of harmonic properties; for if we write x = 0, y = 0; x + Py = 0, x — ay = 0; x — (3y = 0, x + ay — 0, * Consider x = 0, x * >/3y = 0, x + iy = 0. Let X = x + V%, Y = x — V3?/; then x = 0, x + iy = are transformed respectively into X + Y = 0, X — «F = 0; and the cross-ratio of the four lines X = 0, 7 = 0, Z + Y = 0, Z — uY = is — w 2, where w 3 = 1. Similarly for the other combinations.
266 Turner : Plane Cubics ivith Inflexions.
...and impose the condition that x = 0, x + (3y = be harmonic with respect to x — fiy = 0, x + ay = 0, we find /3 = 3a ; and the condition that x + 3a?/ = 0, x — ay = be harmonic with respect to x ± iy = shows that 3a 2 = 1.
Similar statements hold for the lines through B; and also for the set through A, except that in this case the involution is hyperbolic. Con- sequently the sides of the three quadrangles, and hence the vertices, are uniquely determined from the sides of the base quadrangle and its diagonal triangle by means of harmonic properties.
What to read after Plane Cubics With a Given Quadrangle of Inflexions?
You can find similar books in the "Read Also" column, or choose other free books by Bird Margaret Turner to read onlineMoreLess
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