### The Poles of a Right Line with Respect to a Curve of Order *n*

University of Pennsylvania, 1901

Summary

The general subject of poles and polars with respect to Higher Plane Curves has been studied by numerous mathematicians, notably by Steiner, Cremona and Clebsch. Steiner gave in *Crelle's Journal*, Vol. XLVII., a large number of theorems relating to this subject, but he omitted the proofs. They were all proved subsequently by Cremona. Cremona's method was peculiar to himself, that is, he adapted a somewhat more general theory, that of the loci of harmonic means, to the theory of poles and polars. In discussing these problems Miss Vivian uses the analytic method. The particular line of discussion which she has taken up is one which has not been treated in any detail by any former writer. She has handled the subject ably, and has arrived at some very interesting results. In one or two instances her results show that the statements of former writers must be
taken with certain limitations, which do not appear to have been considered. Her principal object is to establish the ways in which the poles of a line are limited when the line has certain prescribed relations to the fundamental curve of the nth order, and to its allied curves, the Hessian and Steinerian. Under particular conditions certain points in the plane will be poles for all lines in the plane, while the other poles, called by the candidate "free poles," vary with the line. Many writers do not class the first as poles at all, but it seems more reasonable to class them with the other poles, since they have all the required properties of such points; and, besides, it is more in keeping with the present tendency of thought on these subjects to do so. The subdivisions of the paper are as follows :

- The pencil of curves of which the poles are base points.
- The related curves.
- Poles when the curve
*u-o* has no singularities.
- The inflection locus.
- Poles when the curve
*u-o* has double points and cusps.
- Intersections of higher order with the Steinerian.
*u-o* with triple points and higher multiple points.

Source: Professor Crawley, "Presentation before the Faculty of Candidates for the Doctorate at the University of Pennsylvania,"
Science, New Series, Vol. 14, No. 348. (Aug. 30, 1901), 338.

The entire thesis is available from the University of Michigan Historical Math Collection:
Vivian PhD Thesis