OVA

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OVA

of curvature of any other curve at a particular or specifiedpoint. See Curve .

OVA (from the Latin ovum, an egg) an ornament in formof an egg, usually employed in the echinus.

OVAL , a figure in geometry, bounded by a curve linereturning to itself.

Under this general definition of an oval is included theellipsis, which is a mathematical oval ; also all other figureswhich resemble the ellipsis, though with very different pro-perties ; and, in short, all curves which return to themselves,go under the name of ovals.

For a description of the mathematical oval, the reader willturn to the article Ellipsis, where, it is presumed, he willmeet with full satisfaction.

One of the most remarkable properties of the oval kind isthe following :

Plate I. Figure 1.—Let c E F G be a circle, o its centre ;draw any line, ei, through the centre o; then take any point,f, in the circumference. Let fi be an inflexible line, and letm be a given point in the line f i ; then, if the point f beconceived to move round the circumference of the circle,while the point, I the end of this line, f i, moves or slidesalong the line e i, the point m will describe an oval, almostsimilar to the conic ellipsis. As we have not seen anyequation of this figure, it is presumed that the followinginvestigation, by the author, will be acceptable :

Draw f h perpendicular to the diameter, e g, of the circle,cutting eg at h ; also draw m p perpendicular to e i, cuttingE i always in p, wherever the point m is situated. Let a bethe point in the straight line e i, in which m will coincidewhen f is brought to e, and b the point where 11 will coincidewhen f comes to g.

Then let a p = x

pm = y

E H =

H F = S11=61IF=i

and eg =z d

From the property of the circle we have h f» = d v — v 1 ;then, by similar triangles, if h and imp, we haveif’:im ! ::hf’:pm ! ;

That is, 5 3 : a 3 : : dv — v 3 ; y>=^-(dv —® s )

Therefore, y = ^(dv — u 3 )*

Then, to find the value of x =. A P, we have

lP a = IM* — P m 3 = a 3 — ^• (dv — u 3 ) = — dv J r v '^

b % b

Therefore, ip = ^-(6 3 — d v + t> 3 ) 2 ;

But im:ip::mf:phj |

That is, a : — dv + i> 3 ) 2 :: c : p H

1 a-\- e q-

IE = IP+ PH+ HE=(i 3 — dv + » 3 ) 2 XI A = I E-A E = —X (6 J — dV + 0 2 ) 2 + * """

ap — i a — ip ■=^ r ( b i — dv v 3 ) ? + v —e ; by ^‘ c ,b J

a /Ji;—’

value of AP, corresponding to pm or y = -j \ a

may be found in the most simple manner. . jo ^

Therefore, if a p in the figure were alwaysversed sine e h of the circle, the curve _descnb^ be >emotion of the point m, would really be an ellipsis, an

p m or y = - (dv — f 3 ) 2 it follows, that the axis P

th® ^ .

dicular to the ordinates, is to the axis parallel to ^ tg jnates, in the ratio of b to a ; that is, in the ratio onearly. - vV jll $

Let a = 20, b = 40, e = 20, and d rs 10 » ^

= —d (v — v 1 ) 2 — — (6 v — d 3 ) 2 ; from which the

b 2 1 . pj «*

values are obtained, according to the different assU®Pthe versed sine, v, of the circle :

i

Assume v z = 1, then (6 X 1 — l 3 )* = (6 — 1)2 = ~ 1*118 ;

£4 Z 2 t

v = 2, I (6 X 2 — 2 3 ) 2 = I (12 — 4) 2 = |- 2 = 1.4142 ;

A A A

« = 3, 1(6 x 3-3 3 )* = l(18-9) 2 =^= 1.5 ;

v = 4, i (6 X 4 — 4 3 )* = A -(24 - 16)* = = 1.4142;

v = 5, l (6 x 5 — 5 3 )* = j (30 - 25)* = = 1.118.

Then, because x = ~ (£ 3 — dv + »•)* + v — c = \ (^ 00 — 6 v -f v 3 ) + v — 20, we shall have the

of x, by the different assumptions of v, which must be those answering to y, as before •

folloff ing

v = 1, then x = 1(1600—6 X 1 + P)* + 1 — 20 = 0.96872 ;

A

9 = 2 , x = 1600 — 6 X 2 + 2*)* + 2 — 20 = 1.94993;

A