CON 
CON 
423 
gm- curve is an' algebraic expression, which 
denotes' the relation betwixt the ordinate and 
abscissa; the abscissa being equal to a, aucl 
the ordinate equal to ?/. 
If p be the parameter of a parabola, then 
<v 2 = fix-, which is an equation for all parabolas, 
be the diameter of an ellipse, p its para- 
If 
P 
meter ; then y 2 — — • — ax -j- xx ; an equation 
for all ellipses. 
If « be a transverse diameter of a hyperbola, 
p its parameter ; then y 2 = — x ax 4- xx. 
1 a 
If a be a second diameter of a hyperbola, then 
P 
y ~ — - X aa xx ; which are equations for 
all hyperbolas. 
As all these equations are expressed by the 
second powers of x and y, all conic sections 
are curves of the second order; and con- 
versely, the locus of every quadratic equation 
is a conic section, and is a parabola, ellipse, 
or hyperbola, according as the form of the 
equation corresponds with the above ones, 
or with some other deduced from lines drawn 
in a different manner with respect to the sec- 
tion. 
General properties of conic sections. — A 
tangent to a parabola bisects the angle con- 
tained by the lines drawn to the focus and 
-directrix; in an ellipse and hyperbola, it bi- 
sects the angle contained by the lines drawn 
, to the foci. 
In all the sections, lines parallel to the tan- 
gent are ordinates to the diameter passing 
through the point of contact ; and in the el- 
lipse and hyperbola, the diameters parallel to 
the tangent, and those, passing through the 
points of contact, are mutually conjugate to 
each other. If an ordinate be drawn from a 
point to a diameter, and a tangent from the 
same point which meets the diameter pro- 
duced ; in the parabola, the part of the dia- 
meter betwixt the ordinate and tangent will 
be bisected in the vertex; and in the ellipse 
and hyperbola, the semidiameter will be a 
mean proportion betwixt the segments of the 
diameter, betwixt the centre and ordinate, 
and betwixt the centre and. tangent. 
The parallelograms formed by tangents 
drawn through the vertices of any conjugate 
diameters, in the same ellipse or hyperbola, 
will be equal to each other. 
Properties peculiar to the hyperbola. — As 
the hyperbola has some curious properties 
arising from its asymptotes which appear at 
first view almost incredible, we shall briefly 
demonstrate them. 
(1) The hyperbola and its asymptotes 
never meet: if not, let them meet in S (fig. 
16.); then by the property of the curve the 
rectangle KX x XE is to SX 2 as GE 2 to 
GM 2 or EP 2 ; that is, as GX 2 to SX 2 ; where- 
fore KX x XE will be equal to the square 
of GX ; but the rectangle KX x XE, toge- 
ther with the square of GE, is also equal to 
the square of GX ; which is absurd. 
(2) If a line be drawn through an hyperbola 
parallel to its second axis, the rectangle, by 
the segments of that line, betwixt the point in 
the hyperbola and the asymptotes, will be 
equal to the square of the second axis. 
For if SZ (fig. 16.) be drawn perpendicu- 
lar to the second axis, by the property of the 
curve, the square of MG, that is, the square 
of PE, is to the square of GE, as the squares 
ZG and the square of MG together, to the 
CON 
square of SZ orGX; and the squares of TTX 
and GX are in the same proportion, because 
the triangles RXG, PEG, are equiangular; 
therefore the squares ZG and MG are equal 
to the square of RX; from which, taking the 
equal squares of SX and ZG, there remains 
the rectangle 11SV, equal to the square of 
MG. 
(3) Hence, if right lines be drawn parallel to 
the second axis, cutting an hyperbola and its 
asymptotes, the rectangles contained betwixt 
the hyperbola and points where the lines j 
cut the asymptotes will be equal to each 
other ; for they are severally equal to the 
square of the second axis. 
(4) If from any points, d and S, in an hy- 
perbola, there be drawn lines parallel to the 
asymptotes d a SQ and S b d c, the rectangle 
under d a and d c will be equal to the rec- 
tangle under QS and S b ; also the parallelo- 
grams d a G c, and SQG b, which are equi- 
angular, and consequently proportional to the 
rectangles, are equal. 
For draw Y W RV parallel to the second 
axis, the rectangle Y d x d W is equal to 
the rectangle KS x SW ; wherefore WD is 
to SV as ES is to d Y. But because the tri- 
angles RQS, a YD, and GSV, c d W, are 
equiangular, W d is to SV as c d to S b, and 
RS is to D Y as SQ to d a ; wherefore, d c 
is to S b as SQ to d a ; and the rectangle d c, 
d a, is equal to the rectangle QS, S b. 
(5) The asymptotes always approach near- 
er the hyperbola. 
For, because the rectangle under SQ and 
S b or QG, is equal to the rectangle under 
d a and d c, or AC and QG is greater than 
a G ; therefore, d a is greater than QS. 
(6) The asymptotes come nearer the hy- 
perbola than any assignable distance. 
Let X be any small line. Take any point, 
as d, in the hyperbola, and draw da, d c, 
parallel to the asymptotes: and as X is to 
d a, so let a G be to GQ. Draw QS 
parallel to a d, meeting the hyperbola in S ; 
then QS will be equal to X. For the rec- 
tangle SQ x QG will be equal to the rec- 
tangle d ax a G ; and consequently SQ is 
to d a as A G to GQ. 
If any point be taken in the asymptote be- 
low Q, it can easily be shown that its distance 
is less than the line X. 
Uses of conic sections. — Any body, pro- 
jected from the surface of the earth, describes 
a parabola, to which the direction wherein it 
is projected is a tangent : and the distance of 
the directrix is equal to the height from which 
a body must fall to acquire the velocity 
with which it is projected: hence the proper- 
ties of the parabola are the foundation of gun- 
nery. 
All bodies acted on by a central force 
which decreases as the square of the distances 
increases, and impressed with any projectile 
motion making any angle with the direction 
of the central force, must describe parabolas, 
ellipses, and hyperbolas, according to the 
proportion between the central and projectile 
force. This is proved by direct demonstration. 
The great principle of gravitation acts in 
this manner: and all the heavenly bodies 
describe conic sections, having the sun in one 
of years ; or whether they describe parabqla* 
and hyperbolas, in which case they will nhve* 
return. 
They are of great use also in many other 
parts of the mathematics; in dialing, for de- 
lineating the signs in the projection ot the 
sphere, many of whose circles are projected 
into these curves ; in optics, to reflect or re- 
fract rays accurately to a focus; in loga- 
rithms, and in the higher parts of algebra, one 
or other of them is continually applied ; and 
no one can make any great progress in 
the mathematics, without understanding 
thoroughly the chief properties of conic 
sections. 
The principal writers on conic sections in 
modern days are, Hamilton, Robertson, 
Vince, New ton, &c. 
CONJUGATE diameter, or' axis of an 
ellipsis, the shortest of the tw-o diameters, or 
that bisecting the transverse axis. 
Conjugate hyperbolas. See Conic 
Sections. 
CONJUGATION, in grammar, a regular 
distribution of the several inflections of verbs 
in their different voices, moods, tenses, num- 
bers, and persons, so as t© distinguish them 
from one another. 
The Latins have four conjugations, distin- 
guished by the terminations of the infinitive 
are, ere, ere, and ire ; the vowels before re 
of the infinitive in the first, second, and fourth, 
conjugations, being long vowels, and that be- 
fore re in the infinitive of the third being a 
short one. See Vowel. 
ITe English have scarcely any natural in- 
flexions, deriving all their variations from 
additional particles, pronouns, &c. whence 
there is scarcely any such thing as strict con- 
jugation in that language. 
CONIUM, hemlock, a genus of the digy- 
nia order, in the pentandria class of plants, 
and in the natural n'lethod ranking under the 
45th order, umbellate. The partial invo-' 
lucra are halved, and mostly triphyllous ; the 
fruit subglobose and quinque-striated, the 
stria 1 crenated on each side. There are five 
species ; the most remarkable are : 
1. Conium Africanum, w T ith prickly seeds, 
a native ol the Cape of Good Hope, and rarely 
growing above nine inches high: the lower 
leaves are divided like those of the small 
w'ild rue, and are of a greyish colour ; those 
upon the stalk are narrower, but of the same 
colour: these are terminated by umbels of 
white flowers, each of the larger umbels be- 
ing composed of three small ones ; the invo- 
lucrum has three narrow leaves situated un- 
der the umbel. 
2. Conium maculatum, or the greater hem- 
lock, grows naturally on the sides of banks 
and roads in many parts of Britain. It is a 
biennial plant, which perishes after it has 
ripened its seeds. It has a long taper root 
like a parsnip, but smaller. The stalk is 
smooth, spotted with purple ; and rises from 
four to upwards of six feet high, branching 
out towards the top, with decompounded 
leaves. The stalks are terminated by um- 
bels of white flowers. This species is some- 
times applied externally in the form of de- 
coction, infusion, or poultice, as a discutient. 
of the foci ; the orbits of the planets are el- | These are apt to excoriate, and their vapour 
lipses, whose transverse and lesser diameters is to some particularly disagreeable and 
are nearly equal. It is uncertain whether some hurtful. The stalks are insignificant, and the 
of the comets describe ellipses with very un- roots very virulent. With regard to its vir- 
equal axes, and so return after a great number tue, when taken internally, it has been gene- 
