of the planes BEnG, pAon, and PNO, pno that of the plane pAon, with the planes CPDON, EpGoF respectively. Because the planes pДon, CPDON are perpendicular to the plane BEnG; PNO must be perpendicular to the plane BEn G (Euc. B. XI. p. 19.) and consequently perpendicular to the two lines AN, ND drawn in that plane (Euc. B. XI. Def. 3.); for the same reason pno is perpendicular to the two lines An, n G. Hence by the property of the circle CNx ND PN' or PN'

ND=

; and En x n G=pn2, or nG= CN

=

pn2

En

Now since An is parallel to BE, and CD parallel to EG, the figure CNnE is a parallelogram, .. CN= En. By similar triangles AND, An G, AN: An :: ND : PN2 :: (since CN= En) PN': pn'. CN En

nG ::

:

pn2

(2.) Hence the nature of the curve APp is such, that if it begins to be generated from the given point A, and PN is drawn always at right angles to AN, AN will vary as PN'. And the same may be said with respect to the relation of AN and NO on the other side of ANn.

(3.) Next, let the plane MPAM be drawn, as before, perpendicular to the plane BEG, but passing through the sides of the cone BE, BG; then the curve MPAOM, formed by the intersection of this plane with the surface of the cone, is called an Ellipse. page 4.)

(Fig. in

In this case, draw two planes, CPDON, HpKon, parallel to the base of the cone; then, for the same reason

as before, PN will be perpendicular both to AN and ND, and pn will be perpendicular both to An and nK; .. NC x ND = P N', and nHx nK= pn'.

By sim. triangles AND, AnK; MNC, MnH, we have AN:

An

:: ND

: nK,

NM : n M

::

NC

: nH;

.: AN x NM: An x nM :: NC x ND: nHxnK

(4.) The nature of the curve APM therefore is such, that if A and M are given points, and PN be always drawn at right angles to AM between the points A, M, AN× NM will vary as PN; and the same with respect to the relation between ANx NM and NO'.

(5.) Lastly,

(5.) Lastly, let the plane pДon be drawn, as before,

perpendicular to the plane BEG, but cutting the side BE in A, and, when produced, meeting a plane drawn touching the other side GB produced, in M; then the curve pPAOo formed by the intersection of the plane pДon with the surface of the cone, is called an Hyperbola.

Let the plane CPDON be drawn parallel to the base; then, by similar tri

NM

n M

::

ND :

nG;

.. AN× NM: AnxnM :: ND× NC: nGxnE

:: PN' : pn2.

(6.) Hence the nature of the curve APp is such, that if A and M are given points, and PN be always drawn at right angles to AN, the point A lying between M and N,

then

then ANX NM will vary as PN'; and the same with respect to the relation of ANx NM and NO'.

II.

Having thus explained the nature of the curves arising from the intersection of a plane with the surface of a cone, we now proceed to shew how these curves may be constructed geometrically.

(7.) Let ELF be a line given in position, and LZ another line drawn at right angles to it in the point L.

In LZ take any point S, and bisect SL in A. Let a point P move from A, in such a manner that it may always be at equal distances from S and the line ELF (or, in other words, let the line SP revolve round S as a center, and intersect another line PM moving parallel to LZ, in such a manner that SP may be always equal to PM); then the point P will trace out a curve O AP, having two similar branches AP, AO, one on each side of the line AZ; which curve will be a Parabola.

To

To shew that this curve will be a parabola, draw PNO at right angles to AZ; then LNPM will be a parallelogram, and LN=PM=SP; but LN AN+AL =AN+AS (since AL-AS by construction), .. SP= AN+AS.

Since 4 a is a constant quantity, x varies as y2, or AN∞ PN'; the relation between AN and PN is therefore the same as in Art. 2.; hence the curve AP is a Parabola.

(8.) Next, (Fig. in page 8) take any line SH, and produce it both ways towards A and M. Let a point P begin to move from A, in such a manner that the sum of its distances from S and H may be always the same (or, in other words, let two lines, SP, PH, intersecting each other in P, revolve round the fixed points S and H, in such manner that SP+ PH may be a constant quantity); then the curve AP MO traced out by the point P will be an Ellipse.

To prove this, it may be observed that when P is at A, then HA+AS or HS+2AS, is equal to that constant quantity; and when P is at M, SM+MH or HS+ 2 HM, is equal to the same quantity. Hence HS+2AS =HS+2HM, from which it appears that 24S=2HM,

or