101. COR. 8. If a tangent be drawn as before, OR. OTA02, and OR'. OT'=OA'2:

102. COR. 9. If a tangent be drawn as before,

103. DEF. 18. The space on any diameter of a curve between the points at which that diameter, produced if necessary, is intersected by a tangent, and an ordinate drawn from the point of contact, is called a subtangent.

PROPOSITION XX.

104. If a tangent be drawn at any point in the curve of a parabola so as to cut any diameter produced, also if an ordinate to that diameter be drawn from the point of contact, the distance on the diameter, from the ordinate to the extremity of the diameter, is equal to the distance from that extremity to the tangent.

Let P'T be a tangent to the curve, at P', meeting the diameter BA produced in T; also, the P co-ordinate axes coinciding with that diameter, and with a tangent AY at A, let P'R be an ordinate drawn from P'; then

B

It is manifest that, if P'R were produced to Q, since AR remains the same, and RP'=RQ, a tangent drawn from Q would meet BA produced, in T.

105. COR. 1. If any number of parabolas be formed on a common axis, EC, and if points D, D', D", &c. in the curves be taken, in the direction of the same ordinate HD to that axis; the straight lines touching the curves at those points will all meet in one point T' on the same axis. For, by the proposition, in all the parabolas,

HE=ET';

and HE is constant; therefore ET' is constant.

106. COR. 2. In all the parabolas, on the same axis, the several ordinates corresponding to any one abscissa are to one another as the square roots of the parameters.

For y, y', &c. being the ordinates, x the common abscissa, and p, p', &c. the parameters of the axes for the different curves; we have (Art. 72.)

y=p1 x1, y' =p" x3, &c.

in all which equations, x being constant, y varies with p'; therefore, &c.

107. COR. 3. If a tangent be drawn from any point, as D' in a parabola, to meet the axis of the curve produced, as in T'; the distance of the focus from the point of contact is equal to the distance, on the axis, from the focus to the point in which the tangent meets the axis.

Let F be the focus; then FD'FT'.

For p being the parameter of the axis, and EH (=ET') being represented by x,

p + 4 x
4

also FE=2; therefore FT=2+x, or

Thus

4

FD' FT.

It follows that the angle FT'D'F D'T'.

PROPOSITION XXI.

108. A line drawn from the focus of a parabola, to any point in the curve, is equal to one quarter of the parameter of the diameter passing through that point. Let AB be the axis, F the focus, P any point in the curve, and PQ the diameter passing through P; then, if p' be the parameter of PQ,

FP=p'.

Draw the ordinate PR to the axis, and let PT be a tangent at P meeting BA produced in T; also draw AR', an ordinate to PQ, from a the vertex. Let P be the parameter of the axis, and let AR be represented by x: then (Art. 104.) RT= 2x, and (Art. 72.) RP2=px;

therefore, PT2, or R'A2,=px+4x2, or=x (p+4x). But (Euc. 34. I.) R'PAT=(Art. 104.) AR, or x; therefore (Art. 72.),

R'A2=p'x.

Consequently, p'x=(p+4x) x, or p'=p+4x.

FRAT

N

But [(c) Art. 82.] Fr=

p+ x

+4*; therefore FP=\p'.

4

109. COR. If a double ordinate, as MN, of any diameter, as PQ intersecting PQ in R", be equal to the parameter of that diameter, it will pass through F, the focus.

But MN, being a double ordinate of the diameter PQ, is parallel to PT; therefore MN passes through F.

PROPOSITION XXII.

110. If a normal be drawn at any point in the curve of an ellipse, or hyperbola, to cut either of the axes, and if an ordinate be drawn from the point to that axis, the square of the semi-axis on which the normal falls will be to the square of the conjugate semi-axes as the distance of the ordinate from the centre of the curve, on the first axis, is to the distance between the ordinate and the normal.

Let P'NN', NP'N', be, respectively, an ordinate to an ellipse and an hyperbola at the point P', meeting the axes EC, Zz' in N and N', and let the ordinates P'R, P'R' be drawn to those axes: then

and

OE20Z2:: ORRN,
OZ2: OE2: OR': R'N'.

The co-ordinate axes coinciding with OE and Oz, let x', y' be co-ordinates of P', and x, y those of any point in the normal P'N; also representing OE and Oz, respectively, by t and c: then the equation for a normal to an ellipse will be [(d') Art. 92.]

but, at the point N, where y=0, and x=ON, the equation becomes

In the equation for the normal making a=0, in which case -y=ON', that equation becomes

or

whence we have

t2y'x' = c2x'y' — c2x' Y,

t2

y' = y'―y ;

c2 t2:: y' (=OR′) : y'—y (=OR'+ON')=R'n'.

The equation for a normal to an hyperbola is [ (e') Art. 92.] 'x — t2y'x' = c2 x'y' — c2x'y,

t2y'x

111. DEF. 19. The part of either axis of an ellipse or hyperbola between the points at which the axis is intersected by a normal drawn from any point in the curve, and by an ordinate to the axis, drawn from the same point, is called a subnormal.

112. COR. 1. A tangent and a normal being drawn from any point in an ellipse or hyperbola, to meet the transverse axis of the curve in points, as T and N, we have

ON. OT=OE2 Foz2 (=(Art. 69.) the square of the excentricity.) [Note. The upper sign is for the ellipse, and the lower for the hyperbola.]

For, t and c being put for the semi-transverse and semiconjugate axes, respectively, of either curve, and e for the excentricity,

113. COR. 2. A normal being drawn from any point P' in an ellipse or hyperbola to cut the transverse and conjugate axes of the curve in points, as N and N';

but, from the similarity of the triangles P'R'N', NON',

114. COR. 3. A tangent and normal being drawn as in the first and second corollaries;

NR. OT=072, and N'R'. OT'=OE2.