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III. — A new Discussion of the General Equation of Curves 

 of the Second Degree. 



By Mr. Robert Finlay, Professor of Mathematics, 

 New College, Manchester. 



[Read December 2nd, 1851.] 



The method of representing the position of a point in a plane 

 is the fundamental principle of the Cartesian Geometry. 

 To understand this method, let Ox and Oy (Fig. 1) be two 

 straight lines cutting each other in O, and P any point in the 

 same plane. Draw PQ and PR parallel to Ox and Oy; 

 then, if the lengths of PQ and PR be given, the position of 

 P in reference to Ox and Oy may be considered as known. 

 Hence PQ. and PR are called the co-ordinates of P, and they 

 are denoted respectively by x and y. The lines Ox and Oy 

 are called the axes of co-ordinates, and are always either 

 given or assumed. The point O is called the origin of 

 co-ordinates. 



If this method of determining the position of a point be 

 viewed in connexion with the well known algebraical fact, 

 that any equation containing two variables, such as 



2 x"" + S xy = 5, 

 can be satisfied in an infinite number of ways, by assigning 

 particular values to x and corresponding ones to y, it will 

 readily appear that any algebraical equation containing two 



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