cular let fall from the intersection of two tangents which cut at right angles on their chord of contact ;" and "The parameter of a conic is found by dividing four times the rectangle under the segments of a focal chord by the length of that chord" (Art. 193, Ex. 1).

Ex. 3. If a and b be the lengths of two tangents to a parabola which intersect at right angles, and m one quarter of the parameter, prove

206. If in the original equation gẞ=fa, the coefficient of x vanishes in the equation transformed as in the last article; and that equation by2 + 2ƒ'y + c' = 0, being equivalent to one of the form b' (y-λ) (y — μ) = 0,

represents two real, coincident, or imaginary lines parallel to the new axis of x.

We can verify that in this case the general condition that the equation should represent right lines is fulfilled. For this condition may be written

But if we substitute for a, h, b, respectively, a2, aß, ß3, the lefthand side of the equation vanishes, and the right-hand side becomes (fa-gß). Writing the condition fa=gẞ in either of the forms fa gaß, faß = gẞ", we see that the general equation of the second degree represents two parallel right lines when h2 = ab, and also either af=hg, or fh=bg.

=

*207. If the original axes were oblique, the equation is still reduced, as in Art. 205, by taking for our new axes the line ax +ẞy, and the line perpendicular to it, whose equation is (Art. 26) (B-a cos w) x − (a− B cosw) y = 0.

=

And if we write ya2 + B-2aß cos w, the formulæ of transformation become, by Art. 34,

yY=(ax+ẞy) sin w, yX= (B-a cos w) x (α-ẞ cos w) y;

whence

yx sin w = (α- B cos w) Y+BX sin w;

sin w = (B-a cosw) Y-ax sin w. Making these substitutions, the equation becomes

y*Y*+ 2 sin' w (gB – fa) X

+2 sin w {g (a-ẞ cos w) + ƒ (B-a cos w)} Y + yc sin3w = 0.

And the transformation to parallel axes proceeds as in Art. 205. The principal parameter is

208. From the equation y=px we can at once perceive the figure of the curve. It must be symmetrical on both sides of the axis of x, since every value for x gives two equal and opposite for y. None of it can lie on the negative side of the origin, since if we make a negative, y will be imaginary, and as we give increasing positive values to x, we obtain increasing values for y. Hence the figure of the curve is

that here represented.

F

M

Although the parabola resembles the hyperbola in having infinite branches, yet there is an important difference between the nature of the infinite branches of the two curves. Those of the hyperbola, we saw, tend ultimately to coincide with two diverging right lines; but this is not true for the parabola, since, if we seek the points where any right line (x=ky+1) meets the parabola (y2 = px), we obtain the quadratic

y2 - pky - pl = 0,

whose roots can never be infinite as long as k and I are finite. There is no finite right line which meets the parabola in two coincident points at infinity; for any diameter (y = m), which meets the curve once at infinity (Art. 142), meets it once also in

2

the point x= ; and although this value increases as m in

Р

creases, yet it will never become infinite as long as m is finite.

209. The figure of the parabola may be more clearly conceived from the following theorem: If we suppose one vertex

and focus of an ellipse given, while its axis major increases without limit, the curve will ultimately become a parabola.

We wish to express b in terms of the distance VF(=m), which we suppose fixed. We have m = a - √√(a2 – b2) (Art. 182), whence b2am - m2, and the equation becomes

Now, if we suppose a to become infinite, all but the first term of the right-hand side of the equation will vanish, and the equation becomes

the equation of a parabola.

y2 = 4mx,

A parabola may also be considered as an ellipse whose eccen

tricity is equal to 1. For e2 = 1 - Now we saw that

b2

a2

which is the coefficient of x2 in the preceding equation, vanished as we supposed a increased, according to the prescribed conditions; hence e becomes finally = 1.

THE TANGENT.

210. The equation of the chord joining two points on the curve is (Art. 86) (y - y') (y — y′′) = y2 — px,

or

(y' + y") y = px + y'y".

And if we make y" = y', and for y" write its equal px', we have the equation of the tangent

2y'y = p(x + x').

If in this equation we put y=0, we get x=-x', hence TM (see fig. next page), which is called the Subtangent, is bisected.

at the vertex.

These results hold equally if the axes of coordinates are oblique; that is to say, if the axes are any diameter and the tangent at its vertex, in which case we saw (Art. 203) that the equation of the parabola is still of the form y2=p'x.

from P to any other diameter,

since we have only to take V'M' T'V', and join PM'.

=

211. The equation of the polar of any point x'y' is similar in form to that of the tangent (Art. 89), and is, therefore,

2y'y= p(x+x').

Putting y=0, we find that the intercept made by this polar on the axis of x is-x. Hence the intercept which the polars of any two points cut off on the axis is equal to the intercept between perpendiculars from those points on that axis; each of these quantities being equal to (x − x').

DIAMETERS.

212. We have said that if we take for axes any diameter and the tangent at its extremity, the equation will be of the form y=p'x.

We shall prove this again by actual transformation of the equation referred to rectangular axes (y2=px), because it is desirable to express the new p' in terms of the old

p.

If we transform the equation y" px to parallel axes through any point (x'y') on the curve, writing x + x and y+y' for x and y, the equation becomes

y2+2yy' = px.

Now if, preserving our axis of x, we take a new axis of y, inclined to that of x at an angle 0, we must substitute (Art. 9), y sine for y, and x+y cose for x, and our equation becomes. y' sin'0+2y'y siu 0=px+py cos 0.

In order that this should reduce to the form y=px, we must have

Now this is the very angle which the tangent makes with the axis of x, as we see from the equation

2y'y = p(x + x).

DD

The equation, therefore, referred to a diameter and tangent, I will take the form

The quantity p' is called the parameter corresponding to the diameter V'M', and we see that the parameter of any diameter is inversely proportional to the square of the sine of the angle which

its ordinates make with the axis, since p

=

sin 0

We can express the parameter of any diameter in terms of the coordinates of its vertex, from the equation tan = 2; hence

Р

2y

213. The equation of a line through (x'y') perpendicular to the tangent 2yy' = p(x+x') is

P

MN (the subnormal, Art. 181) = {p.

Hence in the parabola the subnormal is constant, and equal to the semi-parameter. The normal itself

= √(PM2 + MN2) = √(y'* + }p2) = √ {p (x2 + }p}} = {√(pp ́).

THE FOCUS.

214. A point situated on the axis of a parabola, at a distance from the vertex equal to one-fourth of the principal parameter, is called the focus of the curve. This is the point which, Art. 209, has led us to expect to find analogous to the focus of an ellipse; and we shall show, in the present section, that a parabola may in every respect be considered as an ellipse, having one of its foci at this distance and the other at infinity.