NOTE ON TRILINEARS. 251 



and since, by the property of homogeneous functions, 



the equation finally reduces to 



da ^ '^ dj3 ^ ^ dy 



4. The polar of a point. 



Let (a, 13, y) be the point ; (a,, /?j, 7 J, (o.^, P^-,y^) the points of 

 contact of the two (real or imaginary) tangents drawn from^^it^to the 

 conic. 



The equations to these tangents are 



da^ d/S^ dy^ 



and since (a, /8, y) lies in each of these, we have the relations, 



aa, dfii rfy, 



da^ dj3^ dy^ 



and these, by the property of homogeneous functions, are equivfi" 

 lent to 



^4> , n d4» d(fi 



°» ^a + ^^ 5^ + -y^ Ty = ^' 



d^ r. dffi d^ 



""^da-^^^dp-^ y^ dy^"^' 

 and the points (a^, jS^, y^), (ag, jS^, yg) therefore lie iii the line 



«' r + -8' ^t + v' ? = »• 



a a ap dy 



which is therefore the equation to the polar of (a, j8, y). 



CoH, Since the centre is the pole of the line at infinity, the equation 



„'5^+ /S-^ + y ^=0 

 da. dp dy 



