290 A. Mukbopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



to this method, there are two points which ought to be most carefully 

 noticed. In the first place, to determine the position of any point, 

 we must choose an origin, and, then, fix the position of the point by 

 its coordinates, which may be defined to be independent quantities of 

 the same order which fix the position of a point ; we see, then, that the 

 two essentially distinct ideas of origin and coordinates are fundamental 

 in this theory ; and, if we consider the matter for a moment, we find 

 that the same two ideas are ever present in every system of coordinates 

 that we may choose. Thus, looking to a comparatively modern part of 

 the subject, the theory of Elliptic Coordinates, we see that the posi- 

 tion of any point is determined by the lengths of the semi-axes of the 

 conies which can be drawn through that point confocal to a given conic, 

 called the primitive conic ; here, then, the point-origin of the Cartesian 

 system has been replaced by the fundamental conic, and the ordinate 

 and abscissa have been replaced by the semi-axes of two conies. 

 Hence, we conclude that in every system, we must have an origin j 

 which is, as it were, a unit or symbol of reference, and which may be 

 a point or a conic, or any other figure, according to the system we 

 choose ; and, having fixed our origin, we determine the position of a 

 point by coordinates, which may be lines straight or curved, or any 

 other geometrical figure ; the only essential ideas being those of a 

 symbol of reference, and of the independence of the quantities which 

 fix the position of the point relatively to that origin or symbol of 

 reference. 



Having thus fixed the position of a point, we next consider how 

 to represent a curve. A curve is defined to be an assemblage of points 

 arranged according to a definite law ; the equation of a curve, therefore, 

 is the analytical representation of that geometrical relation which must 

 subsist between the coordinates of a point, in order that that point 

 may be on the given curve. In other words, the equation of a curve 

 may be defined to be the analytical representation of some geometrical 

 property of the curve ; and, as a curve has an infinite number of geo- 

 metrical properties, the question naturally suggests itself whether the 

 analytical representation of each of these properties will give a different 

 equation of the curve. As a matter of fact, we do know that, in what- 

 ever way we may derive the equation of a curve, we are led to equations 

 which are apparently different from each other, but which are really 

 not distinct, and which may all be made to coincide by suitable trans- 

 formations. Indeed, if the reverse had been the case, it would have 

 been manifestly impossible to create a science of analytic geometry ; 

 and the reason why all the equations of a curve are really identical 

 is a simple outcome of the fact that all the innumerable geometrical 



