1887.] A. Mukhopadhyay— JfemoiV on Plane Analytic Geometry. 291 



properties of any curve are dependent on each other : the truth of any 

 one being assumed, the others can be deduced from it as necessary 

 mathematical consequences. We see, therefore, that though a curve 

 has an infinite number of geometrical properties, it can have only one 

 equation, and this accords with the great Law of Nature that, in 

 every natural systeniy there can he only one relation between the component 

 parts. This, then, is the second fact which made possible the very 

 existence of Analytical Geometry. 



From what has been pointed out above, it is evident that the 

 equation of a curve is, as it were, a convenient repository of all theorems 

 connected with it, and all its properties may be established by algebraic 

 transformation of the equation. From this, as well as from the funda- 

 mental relation between analysis and geometry noted above, it is clear 

 that, to every algebraic transformation, there corresponds a geometrical 

 fact, and vice versa. Take, for example, the subject of the transforma- 

 tion of coordinates. We all know that transformation is of two kinds ; 

 it may be a change to new axes, parallel to the old ones, through a new 

 origin, which may conveniently be termed Translation-transforma- 

 tion ; or, again, the transformation may be to new axes, inclined to 

 the old ones, through the old origin, which may be called Rotation- 

 transformation ; if, in any case, both these kinds are combined, we 

 may call it Oompound-transformation ; and from the known alge- 

 braical formulae for compound transformation, it is clear that this 

 geometrical process is nothing but the exact counterpart of the alge- 

 braic process of linear transformation. Similarly, it may be re- 

 marked that the problem of inversion is a case of quadric transforma- 

 tion. 



§§.3—5. The Bight Line. 



§. 3. The Line at Infinity. — The equation of any line being 



2+1=1, 



a 

 where a, h are the intercepts on the co-ordinate axes, the equation of 

 the line which is at an infinite distance from the origin is obtained by 

 substituting herein 



a = & = oo, 

 which gives 1 = 0. 



Without any real change of generality, we may write this 



X = 

 where \ is any constant ; this, then, is the equation of the line at in- 

 finity ; it will be of use in determining the asymptotes of the conic given 

 by the general equation of the second degree (§. 12). 



