THE AXIS AND ITS TANGENT. FOCAL PROPERTIES. 127 
Let Pp intersect tlie axis Ax of the parabola in 0, and let 
QN be the ordinate of Q. Let AN = a, QN = Z>, Pp = c, ON = z ) 
l = the latus-rectum. Then the value of z will be defined by 
the biquadratic equation 
ZV - 4Plz 3 + PI (4a + T) z* + b* (4al - c a ) = 0. 
Newton: Arithmetica Universalis, prob. xiv. 
Section VI. 
Referred to the Axis and its Tangent. Focal Properties. 
1. P\ P", being any two points in a parabola, and 0 the 
point of concourse of the tangents at these points, and S the 
focus: to prove that 
(.poy {.P"oy 
P'S ~ F'S ' 
Let the equations to the tangents at P', P", be 
, m 
, nt 
y = CLX + —, 
° a 
„ . m 
y = a x + — . 
a 
Combining these two equations we obtain, for the coordinates 
of the point 0, 
ic, = -rv7, y 1 = -m .(a+a). 
1 aa 7 aa 
Also, combining each of the equations to the tangents with that 
to the parabola, viz. y l — 4mx, we have, for the coordinates 
of P', P", respectively, 
Hence