1887.] A. Mukhopadhyay — Memoir on Plane Anahjtic Geometry, 293 

 equation of the second degree is the same as that between the lines 



The quantity c', which occurs in this investigation, may be called 

 the point-function of the conic. 



Definition. — The point-function of any curve with respect to 

 any point is the function which is obtained by substituting the coordi- 

 nates of that point in the expression the vanishing of which gives the 

 equation of the curve. It is clear that the point-function with respect 

 to any point on the curve itself is zero, while the point-function with 

 respect to the origin is the absolute term in the equation of the curve. < 



§. 6. Area of a Triangle. — If the general equation of the second 

 degree 



ax^'^2hxy-^hy'^-{-2gx-\-2fy'\-c = (8) 



represents a pair of right lines, to investigate the area of the triangle 

 formed by these two lines with the line 



lx-\-my =zn. (9) 



Remove the origin to the point 



ah-h^' ah^^y' 

 which is the point of intersection of the pair of lines represented by 

 (8). The two equations then become 



ax''-\--lhxy^'by^=(} (10) 



and 



, / , hf-lg\ , / hq-af\ 



or lx-{-myz=:p, .,, (11) 



Ijhf- hg ) -{'m(hg~af) + n(h^ - ah) 

 ^here p = j^^—^ (12) 



Now, suppose that the lines in (10) are made up of the two 



y-m^x=zO, y-m^x = 0, (13), (14) 



so that 



mi-l-mj=-— (15) 



^1 ^2 = -J (16) 



U^-2ah 

 whence mj^^-^-m^^ = — (17) 



The coordinates of the point of intersection of (11) with (13) are 



