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



given by 



p ')n,p 



l-\-mm^ l-^-mm^ 



If, therefore, S, is the length of the line intercepted between the 

 new origin (which is the point of intersection of the pair of lines) and 

 the point of intersection of (11) with (13), we have 



, PjSl±!!hll (18) 



Similarly, if 8j be the length of the line intercepted between the new 

 origin and the point of intersection of (11) with (14), we have 



8,»=^-l(i±!^) (19) 



Hence, from (18) and (19), we get 



J9* j l + (mi«+Wa^) + mi«m^a { 

 8 >S a = 1 _-_L. 



Therefore, substituting for w^, m^ from the system of equations (15), 

 (16), (17), we get 



i?*-v/|4^»-|.(a-6)^| 



But, if <f> be the angle between the lines given by (10), we have 



tan <l>^ 

 whence sin <^ = 



a+6 * 

 2\/h^ - ab 



^ I 4^» + («-&)>» j' 



so that the area of the triangle in question is 

 = ^ §1 §2 sin ^ 



_ p^\/h^-ab 

 " am'' -2hml-^bl^ 



I iw- h) +^(% - «/) +^(^' -ab)y 



Qi^-abY iam^-2hml-\-bl^) 

 by substituting for jp from (12). Hence, finally, using the determinant 

 notation, and altering the sign of n, we have the general 



Theorem. — If the general equation of the second degree 

 a«?* + 2;iiK?/ + %2-f2^a:+2/2/+c = 

 i2epresents a pair of right lines, the area of the triangle formed by this 



