232 A. Mukhopadhj^ay — Memoir on [Nov. 



4. Materials for a literary history of Hindustan. — By G. A. Grier- 

 son, Esq., C. S. 



5. Notes on ancient mounds in the district of Quetta. — By Major 

 J. T. Garwood, R. E. 



6. The mother of Jehangir. — By H. Beveridge, Esq., C. S. 

 These papers will be published in full in the Journal, Part I. 



7. A Memoir on Plane Analytic Geometry. — By Asutosh Mukho- 

 padhyay, M. A., F. R. A. S., F. R. S. E. Communicated by The Hon'ble 

 Mahendralal Sirkar, M. D., C. I. E. 



(Abstract.) 



The object of the author in the present memoir, has been to bring 

 together a number of theorems and methods in Plane Analytic Geo- 

 metry which have accumulated in his hands during his study of that 

 subject ; some of the easier of these propositions have already been 

 given in the author's Lectures on Analytic Geometry, now in course 

 of delivery at the Indian Association for the Cultivation of Science ; 

 a few have been published elsewhere without demonstration ; most of 

 the theorems, however, are here given for the first time. The paper 

 now printed contains the first thirty-two sections of the memoir, which, 

 when completed, will, in addition to the sections now printed, contain 

 theorems on Elliptic Coordinates, Elliptic Inversion, and other ana- 

 logous subjects. The first section is introductory, and contains a 

 statement of the object of the memoir, and a very brief outline of the 

 principal topics discussed. The second section is devoted to a con- 

 sideration of the notions which lie at the basis of analytical geometry ; 

 the relation between analysis and geometry is pointed out, as well as 

 two fundamental ideas which made possible the existence of analytical 

 geometry ; the terms Translation-transformation, Rotation-transforma- 

 tion and Compound-transformation, which are freely used later on, are 

 here explained for the first time. Sections three to five are devoted 

 to the right line. In the third section is obtained the Cartesian equa- 

 tion of the line at infinity, which is used in the theory of asymptotes 

 given in the twelfth section. The fourth section contains a new proof 

 of the condition that the general equation of the second degree may 

 represent a pair of right lines ; this method has the additional advan- 

 tage of furnishing at once the coordinates of the point of intersection 

 of the two lines given b}' the general equation ; the term Point-func- 

 tion is here first used and defined. The fifth section contains an inves- 

 tigation of the area of the triangle formed by any line with a pair of 

 lines given by the general equation of the second degree ; the length 



