If one also removes the second postulate (" a line can be extended indefinitely ") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x < sub > 0 </ sub > of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic.

So the area enclosed by an ellipse is easy to calculate -- it's the lengths of elliptic arcs that are hard.

where again e is the eccentricity and where the function is the complete elliptic integral of the second kind.

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral.

The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

A radial elliptic trajectory is a non-trivial special case of an elliptic orbit, where the ellipse is a line segment.

The other argument can likewise be expressed as, the amplitude, or as or, where and is one of the Jacobian elliptic functions.

The incomplete elliptic integral of the first kind is defined as

This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by Abramowitz and Stegun and that used in the integral tables by Gradshteyn and Ryzhik.

The incomplete elliptic integral of the second kind in trigonometric form is

The incomplete elliptic integral of the third kind is

A relation with the Jacobian elliptic functions is

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic,

Elliptic curve cryptography ( ECC ) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.

For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible.

The primary benefit promised by ECC is a smaller key size, reducing storage and transmission requirements — i. e., that an elliptic curve group could provide the same level of security afforded by an RSA-based system with a large modulus and correspondingly larger key — e. g., a 256bit ECC public key should provide comparable security to a 3072bit RSA public key ( see # Key sizes ).

If one assumes that the average flux did not change between measurements, a mass-distribution curve is obtained which relates the flux of particles larger than a given radius to the inverse 7/2 power of the radius.

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by Af.

In the following paper it is shown that in a certain definite sense, exactly an odd number of squares can be inscribed in every such curve which does not contain an infinite number of inscribed squares.

If the vertex is at Af, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90-degrees about the vertex.

But Af is just the curve Af translated without rotation through a small arc, for Af is always obtained by rotating C through exactly 90-degrees.

The arc is itself a segment of an analytic curve.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

The order of this congruence is Af, since Af secants of a curve of symbol ( B ) on a quadric surface pass through an arbitrary point.

For the lines of any plane, **yp, meeting Q in a conic C, are transformed into the congruence of secants of the curve C' into which C is transformed in the point involution on Q.

Each generator, **yl, of Af is also exceptional, for each is transformed into the entire congruence of secants of the curve into which that generator is transformed by the point involution on Q.

This curve is of symbol Af since it meets **yl, and hence every line of Af in the Af invariant points on **yl and since it obviously meets every line of Af in a single point.

Hence C' is a Af curve on Q.

By ( 1 ), the image of this pencil is a ruled surface of order Af which is met by the plane of the pencil in a curve, C, of order Af.

To avoid this contradiction it is necessary that C be composite, with the secant of **zg and a curve of order Af as components.

We now observe that the case in which **zg is a Af curve on a quadric is impossible if the complex of singular lines consists exclusively of the lines which meet Aj.

This contradicts the preceding observations, and so, under the assumption of this paper, we must reject the possibility that **zg is a Af curve on a quadric surface.

Continuing with the case in which **zg is a Af curve on a quadric Q, we first observe that the second regulus of Q consists precisely of the lines which join the two free intersections of **zg and the planes through any one of the multiple secants.

We assume that average total unit cost in the relevant region of operation is constant with respect to quantity produced ( the average cost curve is horizontal, and therefore is identical with the marginal cost curve ), and is the same for every firm ( and therefore for the industry ).

Clayton is with him, takin him out of the valley.

His wife had said to him: `` Nellie is in love with Clayton Roy.

`` Exterminatin' cow thieves is just a business proposition with me '', he'd blandly announce.

It is also possible, but equally doubtful, that he actually shot down the hundreds of men with which his legend credits him.

Let me pass over the trip to Sante Fe with something of the same speed which made Mrs. Roebuck `` wonduh if the wahtahm speed limit '' ( 35 m.p.h. ) `` is still in ee-faket ''.

One of my virtues or vices is a sort of three-dimensional imagination complete with sound effects and glorious living color.

this is not so, for education offers all kinds of dividends, including how to pull the wool over a husband's eyes while you are having an affair with his wife.

It is nothing you can put your fingers on but the air suddenly fills with a high charge of electricity.

It is Eromonga -- look hard, you can see with your naked eye the wooden scaffolding on the cliff ''.

I clapped the big man with the bleached hair on his shoulder and said heartily, hoping it would make an impression on the women: `` This one is the maku Frayne.

That place is crawling with Bill Doolin and his gang ''.

The woman eyed the youth with the avidity a coin collector might display toward a rare doubloon which is not yet in his collection.

`` What is with this vow jazz ''??

-- liberal considers that the need for a national economy with controls that will assure his conception of social justice is so great that individual and local liberties as well as democratic processes may have to yield before it.

Why, in the first place, call himself a liberal if he is against laissez-faire and favors an authoritarian central government with womb-to-tomb controls over everybody??

In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.

Reduced to its simplest terms, it is an assumption of a collective duty to compensate for the inability of individuals to cope with the rigors of the era.

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

Only one rule prevailed in my conversations with these men: The more highly placed they are -- that is, the more they know -- the more concerned they have become.

He was, and is, with the RAND Corporation, a nonprofit pool of thinkers financed by the U.S. Air Force.

They include the Navy's Atlantic Command at Norfolk, Virginia, which is in contact with the Polaris subs ; ;

In point of fact, this is a beige box with a bright red door, about one and a half feet square and hung from the wall about six feet from the door to Wisman's right.

It has nothing of the proud stride of the trained runner about it, it is not a lope, it is not done with style or verve.