ELL 
the ellipsis, they may be reduced to the fol- 
lowing propositions. 1. If from any point 
M in an ellipsis, two right lines, MF, M f, 
(fig. 1.) be drawn to the foci F ,f, the sum 
of these two lines will be equal to the trans- 
verse axis A B. This is evident from the 
manner of describing an ellipsis. 2. The 
square of half the lesser axis is equal to the 
rectangle under the segments of the greater 
axis, contained between the foci and its. 
vertices ; that is, D C 2 = A F X FB = A/ 
X/B. 3. Every diameter is bisected in 
the centre C. 4. The transverse axis is the 
greatest, and the conjugate axis the least, 
of all diameters. 5. Two diameters, one 
of which is parallel to the tangent in the 
vertex of the other, are conjugate diameters ; 
and vice versa, a right line drawn through 
the vertex of any diameter parallel to its 
conjugate diameter, touches the ellipsis in 
that vertex. 6. If four tangents be drawn 
through the vertices of two conjugate dia- 
meters, the parallelogram contained under 
them will be equal to the parallelogram 
contained under tangents drawn through the 
vertices of any other two conjugate dia- 
meters. 7. If a right line, touching an el- 
lipsis, meet two conjugate diameters pro- 
duced, the rectangle under the segments of 
the tangent, between the point of contact 
and these diameters, will be equal to the 
square of the semi-diameter, which is con- 
jugate to that passing through the point of 
contact. 8. In every ellipsis, the sum of 
the squares of any two conjugate diameters 
is equal to the sum of the squares of the two 
axes. 9. In every ellipsis, the angles F G I, 
/GH, (fig- 1), made by the tangent HI, 
and the lines FG,/G, drawn from the foci 
to the point of contact, are equal to each 
Other. 10. The area of an ellipsis is to the 
area of a circumscribed circle, as the lesser 
axis is to the greater, and vice versa with 
respect to an inscribed circle ; so that it is 
a mean proportional between two circles, 
having the transverse and conjugate axes 
for their diameters. This holds equally 
true of all the other corresponding parts 
belonging to an ellipsis. 
The curve of any ellipsis may be obtain- 
ed by the following series. Suppose the 
semi-transverse axis GB = i', the semi-con- 
jugate axis CD = c, and the semi-ordi- 
nate = a ; then the length of the curve 
. r 2 a? . 4 r 2 c 2 a 6 — r a’’ , 
M B = a + — + 
8 c 4 r 2 a 7 -|- r 6 a 7 — 4 c 2 r 5 a 7 
112 c 1 - 
40 c 8 
See. 
ELL 
this series will be more simple : 
c — 2 r, then MB = «-f 
113 a 7 , 3419 « 9 
«2 1 
for if 
3 a s 
+ 
96 r 2 1 2048 r 4 
&c. This se- 
438752 a 6 “75497472 r 8 ’ 
ries will serve for an hyperbola, by making 
the even parts of all the terms affirmative, 
and the third, fifth, seventh, &c. terms ne- 
gative. 
The periphery of an ellipsis, according to 
Mr. Simpson, is to that of a circle, whose 
diameter is equal to the transverse axis 
t- „• , d 3 d 2 
ot the ellipsis, as 1 — 
And, if 
the species of the ellipsis be determined. 
id 3 
2.2 2.2.4. 
2. 3. 5. 5 .7<f* 
2 . 2 . 4 . 4 . 6 . 6 2 . 2 . 4 . 4 . 6 . 6 . 8 . 8 , 
&c. is to 1, where d is equal to the differ- 
ence of the squares of the axis applied to 
the square of the transverse axis. 
Ellipsis, in grammar, a figure of syn- 
tax, wherein one or more words are not ex- 
pressed ; and from this deficiency it has got 
the name ellipsis. 
Ellipsis, in rhetoric, a figure nearly al- 
lied to preterition, when the orator, through 
transport of passion, passes over many 
things: which, had he been cool, ought to 
have been mentioned. In preterition, the 
omission is designed ; which, in the ellipsis, 
is owing to the vehemence of the speaker’s 
passion, and his tongue not being able to 
keep pace with the emotion of his mind. 
ELLIPTIC, or Elliptical, something 
belonging to an ellipsis. Thus we meet 
with elliptical compasses, elliptic conoid, 
elliptic space, elliptic stairs, &c. The ellip- 
tic space is the area contained within the 
curve of the ellipsis, which is to that of a 
circle described on the transverse axis, as 
the conjugate diameter is to the transverse 
axis ; or it is a mean proportional between 
two circles, described on the conjugate and 
transverse axis. 
ELLIPTOIDES, in geometry, a name 
used by some to denote infinite ellipses, 
:bx 
defined by the equation ay m n - 
(a — x) n ‘ 
Of these there are several sorts: thus, 
if aif—bx 2 (a — x) it is a cubical ellip- 
toid ; and if a y 4 = b x 2 (a — a;) 2 , it de- 
notes a biquadratic elliptoid, which is an 
ellipsis of the third order in respect of the 
appolionian ellipsis. 
ELLISIA, in botany, so called in me- 
mory of John Ellis, F. R. S. a genus of 
the Pentandria Monogynia class and order. 
Natural order of Lurid®. Borragineai, 
Jussieu. Essential character : corolla fun- 
I 
