NOV 871894 



Kansas University Quarterly. 



Vol. III. OCTOBER, 1S94. No. 2. 



On the Hessian, Jacobian, Steinerian, Ete. in 

 Geometry of One Dimension. 



BY HENRY B. NEWSON. 



Analytic Geometry of One Dimension, or Linear Analytic Geometry 

 as it is sometimes called, is only another name for the geometric 

 interpretation of the Theory of Binary Algebraic Forms by means of 

 a group of points on a line. Many mathematicians have called 

 attention to this subject as an independent branch of geometry, but 

 few have developed it to any considerable extent. Cayley's sixth 

 Memoir on Quantics deserves mention as one of the early papers 

 bearing on the subject. The second chapter of Clebsch's Theorie 

 der Binaeren Algel>rarisi/icn Fornicn is devoted to the geometric inter- 

 pretation of binary forms, and the results are freely stated in a 

 geometric form throughout the book. Clebsch-Lindemann's Vorles- 

 ungen neber Geometrie, Vol. I, Kap. Ill, carries the development still 

 farther, but yet leaves it very incomplete. 



Throughout the work of Clebsch the algebraic spirit predominates 

 and the geometric theory is a secondary matter. In the present 

 paper the geometric conception is kept in the foreground and the 

 algebraic operations are employed only as a means for developirrg 

 that conception. In the following pages I have collected and re- 

 stated in a geometric form some well-known theorems, and then 

 proceeded to develope (somewhat after the manner of Salmon's 

 Higher Plane Curves) a brief chapter in linear geometry. I have 

 even stated some of the results as nearly as possible in Salmon's 

 language in order to make clearer the analogy between the theorems 

 in one and two dimensions. A sufficient account of the theory of 

 poles and polars of binary forms, so freely used in the following 

 pages, is to be found in Clebsch-Lindemann's Vorlcsungen, etc.. Vol. 

 I, p. 203. 



(108) KAN. UNIV. QUAE.. VOL. Ill, NO. 3. OCT., 1894. 



