II. On the General Equation of Curves of the Second 



Degree. 



By AUGUSTUS DE MORGAN, 



OF TRINITY COLLEGE, CAMBRIDGE, 

 AND PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF LONDON. 



[Read November 15, 1830.] 



The object of this Paper is to draw attention to some 

 properties of Curves of the Second Degree, by means of which 

 the reduction of their equations from one set of axes to another 

 is materially facilitated. Little, if any, notice of these properties 

 has been taken, nor do I remember to have seen their existence 

 mentioned with the exception of two very limited particular 

 cases, viz. that the sum of the squares of conjugate diameters, 

 and the parallelogram formed by them are constant. 



Suppose that any curve of the second degree is referred to 

 axes which make an angle 6 or that £y = 6. Suppose the ori- 

 gin removed to a point whose co-ordinates are m and n in the 

 directions of x and y, and moreover suppose the directions of 

 the axes to be changed so that x'x = <p and y'x = f, x' and y' 

 being the new co-ordinates; and let x'y = f-<p = ff. Let the 



equations of the curve referred to the first and second systems of 

 axes be 



ay* +bxy +cx 2 +dy +ex +f =o, (i) 



a'y* + b'xfy' + c'x' 1 + d'y + e'x' +/' = o ; (2) 



