The Poles of a Right Line With Respect to a Curve of Order N
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WITH RESPECT TO A CURVE OF ORDER n. 23 §5.
Poles when the Base Curve has Double Points AND Cusps.
A double point on Z7is a pole for every line in the plane and presents several peculiar characteristics. It is a double point on its own first polar curve, and therefore corresponds to itself as a point on the Hessian and the Steinerian. It is a double point on the Hessian ^ which represents always two of the points which can be double points for the pencil of curves belonging to a line through it,... and is therefore a double point on the Steinerian. This is a particular case of the following theorem proved by Henrici ^ by a very elegant analysis : " A point whose first polar has a cusp is a cusp on the Steinefian, and one whose first polar has two double points is a double point on the Steinerian. " ^ The tangents to the Hessian at the double point are the same ^ as those of U, and also are tangents to the Stein- erian, since they are line polars of the corresponding point on the Hessian.
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To quickly assess the difficulty of the text, read a short excerpt:
WITH RESPECT TO A CURVE OF ORDER n. 23 §5.
Poles when the Base Curve has Double Points AND Cusps.
A double point on Z7is a pole for every line in the plane and presents several peculiar characteristics. It is a double point on its own first polar curve, and therefore corresponds to itself as a point on the Hessian and the Steinerian. It is a double point on the Hessian ^ which represents always two of the points which can be double points for the pencil of curves belonging to a line through it,... and is therefore a double point on the Steinerian. This is a particular case of the following theorem proved by Henrici ^ by a very elegant analysis : " A point whose first polar has a cusp is a cusp on the Steinefian, and one whose first polar has two double points is a double point on the Steinerian. " ^ The tangents to the Hessian at the double point are the same ^ as those of U, and also are tangents to the Stein- erian, since they are line polars of the corresponding point on the Hessian.
What to read after The Poles of a Right Line With Respect to a Curve of Order N?
You can find similar books in the "Read Also" column, or choose other free books by Roxana Hayward Vivian to read onlineMoreLess
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