296 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



represents that diagonal of the parallelogram which does not pass 

 through the origin. The area of the triangle formed by this diagonal 

 with the first pair is 



2^ (¥- %) +2/ (hg -af)-c (h^ -ah)]'' 



1 J 



4i(h^ - ah)^ (ap - 2fgh'\'hg^) 

 and that formed with the second pair is 



c '{h'~ahY 



4(/i« - ahy daf - 2fgh^hg'') 

 But, since the discriminant vanishes, it is clear that 



2g(hf--bg)-^2fihg-af)-2c(h»-al)=0 

 af-2fgh-Yhg^^c{ah^h'), 

 Hence, adding the above expressions, the area of the quadrilateral in 

 question is found to be 



1 c 



2 ^hJ^Z^i 



It may be noted that this expression is only apparently independent of 

 /, ^, for the vanishing of the discriminant shews that a, &, c, h are 

 functions of / and g. 



§§. 6—7. The Circle. 

 §. 6. Meaning of the Constants in the Equation of a Circle. — 



The equation of a circle 



x'J^y^^2gx-^2fy^c = 

 being thrown into the form 



the quantities — g, —f are seen to be the coordinates of the centre, while, 

 if r be the radius, we have 



To determine the geometric meaning of c, let 8 be the distance of the 

 centre from the origin, and ^, either of the tangents drawn from the 

 origin to the circle ; then, 



5^ = r'' + ^« 

 and, also, ^'=f-\-g^ 



r'=f^g'-c 



which give c = t^. (25) 



Hence, c denotes the square of the tangent drawn from the origin to the 

 circle. We thas infer that, if the equations of a system of circles agree 

 in either / or ^, the locus of their centres is a right line parallel to a 

 given line at a given distance from it, and their common chords are 

 parallel, being all perpendicular to this given line ; if both / and g are 



