64 ON SOME PROPERTIES OF THE PARABOLA. 



form of the equation to the tangent of the parabola which is 

 given in Art. 2 of our first Number*. 



Let the parabola be referred to its vertex, then the equation 

 to its tangent by that article is 



x 



y = - + *> 



where a is the tangent of the angle which the tangent makes 

 with the axis of y. If a! be the corresponding quantity for 

 another tangent, its equation will be 



x , 



y -7 + ma. . 



Combining these equations, we shall find for the co-ordinates 

 of the point of intersection of the two tangents 



x ma.*, y = m (a + a'). 



We shall distinguish the tangents which form the different 

 sides of the hexagon by suffixing numbers to the a which deter- 

 mines their position, and we shall likewise distinguish the co- 

 ordinates of the summits of the hexagon by suffix letters. 



The equations to the three diagonals are these : 



= m { 2 + a,) a e 6 - (a g 



(3) yfcfr-wJ-xfa + ^-ti-aJ 



= m {(a 3 + aj ai 6 - (a x + a 6 ) a a otj. 



Expressions which, as they ought to be, are symmetrical with 

 respect to the a's. 



Multiply (1) by o e , (2) by - or 4 , (3) by cr 2 , and add. Then y 

 will disappear, and we shall find 



Again, multiply (1) by a,, (2) by -of 1? (3) by 6 , and add: 

 as before, y will disappear, and we shall find the same value for 

 x. Consequently two straight lines whose equations are 



* Cambridge Mathematical Journal, Art. i, No. I. Vol. I. p. 9. 



