S4t 



ON THE GENERAL EQUATION OF CURVES 



variables can represent an infinite number of points ; and it is 

 easy to show that all these points lie in a certain curve, which 

 is called the locus of the equation. Conversely, it is evident 

 that every regular curve must have an equation, and by means 

 of this equation any question relative to the properties of the 

 curve may readily be reduced to algebraic form. Thus, by 

 means of the simple conception of Descartes, the science of 

 geometry was brought within the range of algebraic analysis, 

 and it acquired instantaneously a generality and power which 

 had not been imparted to it by the united efforts of many of 

 the greatest men of antiquity. 



From the statement which I have just made relative to the 

 first principle of the Cartesian Geometry, it will be perceived, 

 that, in that system, the investigation of the properties of any 

 curve depends on the discussion of the equation to the curve. 

 When the equation is of the second degree, its discussion can 

 be effected without much difficulty, and in a tolerably com- 

 plete form. The numerous papers on this subject, however, 

 which have appeared from time to time, and some of which 

 are of very recent date, prove sufficiently that the present 

 mode of discussing the equation of the second degree is not 

 altogether satisfactory ; and this constitutes the most obvious 

 apology which I can offer for introducing to this Society a 

 subject so well known to mathematicians of every degree of 

 attainment. 



The common method of discussing the general equation of 

 the second degree consists mainly in a process called the 

 transformation of co-ordinates. The co-ordinates of a point 

 may be changed in two distinct ways. 1st. By simply 

 changing the origin, the axes remaining parallel to their 

 original directions. 2nd. By turning the axes about the 

 origin into any new positions. The known formulas for the 

 former transformation are so simple and natural that nothing 

 further on that point can be desired. Those for the latter, 

 on the contrary, are exceedingly complicated ; and they have 



