234 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [Nov. 



tween Laplace's Equation and the Theory of Elliptic Motion ; the 

 nineteenth section throws still further light on the matter by a geometric 

 interpretation and a reference to Gauss's Characteristic Equation ; the 

 twentieth section shews how the equation for the eccentricity may be 

 obtained from Laplace's Equation. The twenty-first section primarily 

 deals with the area of the triangle formed by two tangents drawn from 

 any point to a conic and the line joining their points of contact ; the 

 length of the chord of contact is also found ; numerous interesting 

 applications of the formulae are added ; thus, the area of the quadri- 

 lateral formed by two tangents and the two central radii- vectores to 

 the points of contact, is calculated ; and, finally the very interesting 

 theorem is established that any point is outside a conic, on the curve, 

 or inside it, according as the point-function is positive, zero or negative. 

 The next two sections treat of the inclinations of tangents to conies ; 

 the twenty-second section gives a very general theorem connecting the 

 inclinations of any two tangents to a conic and of the chord of contact, 

 to any line, while the twenty-third section gives some geometrical 

 applications, which clearly bring out the correlation between some 

 properties of the circle and the ellipse. The twenty-fourth section 

 furnishes a method of generating similar conies ; the case of the 

 equilateral hyperbola is shewn to be a limiting case, in a very peculiar aud 

 special sense. In the twenty-fifth section, it is proved, as an illustra- 

 tion of the general theory of envelopes, that the envelopes of the sides 

 of an equilateral triangle inscribed in any given triangle, are three 

 parabolas, which are connected by some very neat geometrical relations. 

 The twenty- sixth section deals with the reciprocals of central conies, 

 and, it is shewn that the second focal pedal of a conic is the inverse 

 of a conic. The twenty -seventh section treats of the reciprocal polars 

 of evolutes of a family of curves which include conies as a very 

 particular case ; the formulae are finally extended to the case of any 

 curve, and, it is shewn that if the coordinates of a point on the primitive 

 curve can be expressed by means of a single variable parameter, the 

 coordinates of the corresponding point on the reciprocal polar of the 

 evolute may be similarly expressed ; the analytical theorems obtained 

 in this section are of very great generality, and some of them, of 

 beautiful symmetry. The twenty-eighth section gives various mis- 

 cellaneous properties of the ellipse, while the twenty -ninth section is 

 concerned with two theorems on plane confocals. The next two sections 

 deal with the parabola ; the thirtieth section solves a purely dynamical 

 problem, which is applied in the thirty-first section to obtain some 

 beautiful properties of the parabola, relating to the sum of the squares 

 of the reciprocals of the radii-vectores of the pedals of that curve. 



