CONIC SECTIONS, ANALYTICAL. 153 



straight line, perpendiculars drawn in opposite directions being of 

 course affected with opposite signs. 



The condition that the straight lines 



la + m/3 + ny = 0, to. + m'fi + n'y = 0, 

 shall be parallel is 



mn - m'n + rd'- n'l + lm'-tm = Q; 

 and that they may be perpendicular is 



It sin* A + ... -f ... = (mn' + m'n) sin B sin C cos-4-f- ... + ... 



If I = m = n, or T = m = n, it will be noticed that both these 



conditions are satisfied. The straight line a + /? + y = 0, tfte line at 



'.'/, may then be regarded as both parallel and perpendicular 



to eveiy other straight line ; the angle which it makes with any 



straight line being really indeterminate. 



In questions relating to four given points, it is convenient to 

 take the four points to be 



la = m/3 = ny ; 



and similarly to take the equations of four given straight lines to be 

 la m/J ny - 0. 



The general equation of a conic is 



la* + mfi* + ny' + 2^/?y + 2m'ya + 2'a/2 = ; 

 the polar of any point (a x , /?", /) being 



special forms of this equation most useful are 



(1) circumscribing the triangh nee 



Ifiy -f mya + na/J = ; 



(2) inscribed in the triangle 



(Ja)* + (*!/?)* + (ny)*==0; 



(3) touching two sides at the points where the third meets 

 them 



