288 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



XX. — A Memoir on Plane Analytic Geometry, — By Asutosh Mukhopa- 

 DHYAY, M. A., F. R. A. S., F. R. S. E. Communicated by The 

 Hon'ble Mahendralal Sircar, M. D., C. I. E. 



[Received October 27th ;— Head November 2nd, 1887.] 

 (With three Wood-cuts.) 



Contents. 



§. 1. Introduction ; object and scope of the memoir. 



§. 2. Basis of analytical geometry ; relation between analysis and geometry. 



§§. 3—5. The Right Line : 



(§. 3). The Line at Infinity. 



(§. 4). Coordinates of intersection of two lines given by the ge- 

 neral equation of the second degree : the Point-function, 

 (§. 5). Area of triangle formed by any line with two lines given 

 by the general equation. 

 §§. 6—7. The circle. 



(§. 6). Meaning of the constants in the equation of a circle. 

 (§. 7). Chords and tangents of circles and conies; geometrical 

 meaning of Burn side's Equation. 

 §§. 8 — 15. The general equation of the second degree in Cartesian coordi- 

 nates. 



Preliminary remarks on the general equation. 

 Transformation of the equation ; the Asymptotic Con- 

 stant. 

 Invariants and C ©variants of a single conic. 

 Lengths of axes and area of a conic. 

 Asymptotes of the conic. 

 Equations for the eccentricity of the conic. 

 The Director-circle in rectangular and oblique coordi- 

 nate-axes. 

 (§. 15). Reduction of the general equation to the asymptotes as 

 coordinate-axes. 

 §§. 16 — 20. Laplace's linear equation to a conic. 



(§. 16). Genesis of Laplace's equation. 

 (§. 17). Meaning of the constants. 

 (§. 18). Elliptic motion. 

 (§. 19). Geometric interpretation. 

 (§. 20). Eccentricity from Laplace's equation. 

 §. 21. Area of triangle formed by two tangents to a conic and the chord of 



contact ; various applications. 

 §§. 22 — 23. Theorem on the inclination of tangents. 



(§. 22). Formula for the inclination of tangents to conies and of 



their chord of contact, to any line. 

 (§. 23). Applications of the theorem ; verification. 

 §. 24. Generation of similar conies. 

 §. 25. On a system of three parabolic euYelopes, 



(§. 



8). 



(§. 



9). 



(§. 



10). 



(§. 



11). 



(§. 



12). 



(§. 



13). 



(§. 



14). 



