1887.J A. Muktopadliyay — Memoir on Plane Analytic Geometry. 289 



§§. 26 — 27. Keciprocal polars. 



(§. 26). Reciprocal of central conic. 



(§. 27). Reciprocal of erolute of certain curves including conies. 

 §§. 28 — 29. Theorems on central conies. 



(§. 28). Properties of the ellipse. 



(§. 29). Properties of confocals. 

 §§. 30 — 31. Theorems on the parabola. 



(§. 30). A dynamical problem. 



(§. 31). Applications to the parabola. 

 §. 32. A geometrical locus. 



§. 1. Introduction. 



§.1. Object and Scope.— It is my object in the present 

 paper to bring together a number of theorems in plane analytic geome- 

 try which have accumulated in my hands daring my study of that 

 subject. Some of the simpler of these theorems have already been 

 given in my Lectures on Plane Analytic Geometry, now in course of 

 delivery at the Indian Association for the Cultivation of Science ; 

 a few have been enunciated elsewhere without demonstration ; most 

 of the propositions, however, are here published for the first time. I 

 believe that either the theorems themselves, or the methods of estab- 

 lishing them are original ; and, except in a very few instances where 

 I have inserted well-known results for the sake of avoiding disconnect- 

 edness, I have considered them either for the purpose of giving a proof 

 simpler and more complete than that usually given, or with a view to 

 throw light on the connection between the various parts of the sabject. 

 As the different sections of this paper are, to a great extent, practically 

 independent of each other, for the sake of facility of reference, an 

 outline of the principal topics discussed is added above.* 



§. 2. Basis of Analytical Geometry. 



§.2. Analysis and Geometry.— The notion of either space or 

 number, or of both, lies at the root of every department of mathematics. 

 Analysis is the science of number ; geometry is the science of space ; 

 but, as space is homogeneous, and, as every homogeneous substance 

 can, by the choice of a unit, be represented by a number, space can be, 

 for mathematical purposes, represented by numbers ; hence, the possi- 

 bility of applying analytical methods to geometrical investigations, and 

 of founding a science of analytical geometry. This possibility was 

 first realized into practice by the illustrious French mathematician Hene 

 Descartes, who invented the method of coordinates. With respect 



* For a full analysis of this paper, see the Proceedings for 1887, pp. 232-235. 

 37 



