32 General Meeting for February, 1904. [Feb. 1904] 



papers dealing with Persian and Arabic inscriptions and intended for 

 publication there should therefore be sent to Dr. Ross." 



The following paper was read: — 



The Line at Infitiity. — j5y Indubhushan Brahmachari, M.A. Gom- 

 municated by Mr. C. Little. 



(Abstract.) 



This paper contains a systematic and exhaustive investigation of 

 the properties of what is known to Mathematicians as the Line at 

 Infinity or Line Infinity. The position of any line on a plane may be 

 completely determined, if we know the intercepts which this line makes 

 upon two given intersecting lines which may be taken as the axes of co- 

 ordinates. Now, if these intercepts become infinite in length, the line 

 itself will move off to infinity. In other words, if the equation of the 



line situated within a finite region of the plane be '- + 7 = 1, where a 



a b 



and b are the intercepts, the equation of the line at infinity will be the 

 apparently paradoxical form 1 = or constant = 0. The two fundamental 

 properties of this imaginary line are, first, every point on this line is at 

 infinity, and secondly, that every point at infinity lies on it, or in other 

 words this line is the complete point representative of infinity. Conse- 

 quently, the idea of direction must not be associated with this line. 

 Moreover, it is at the same distance from all ordinary points, because 

 every point of it is at an infinite distance. One of the most familiar 

 instances of the appearance of the line at infinity is in the investigation 

 of the properties of circles, namely it is the imaginary chord of inter- 

 section of all concentric circles. Another instance of its appearance is 

 as the pole of the centre of a conic ; in otiier words, it is the line joining 

 the points of contact of the asymptotes of a hyperbola with the curve. 

 To put the matter in another way, although the asymptote is a tangent 

 whose point of contact is at infinity, it is itself not the line at infinity 

 because it does not lie entirely at infinity. 



The present paper contains a detailed examination of the proper- 

 ties of this line, and shows how its introduction enables us to obtain the 

 solutions of various problems connected with real lines and conies. 

 The paper also contains applications of the properties of this line in 

 connection with the theories of reciprocation and projection. 



