They poured through the opening in the valley, then spread out in a long line to come at us, brandishing their lances and filling the morning with their spine-chilling scalp cry.

Promptly their livestock was taken and according to Gorton the soldiers were ordered to knock down anyone who should utter a word of insolence, and run through anyone who might step out of line.

In his study Samuel Johnson, Joseph Wood Krutch takes this line when he says that what Aristotle really means by his theory of catharsis is that our evil passions may be so purged by the dramatic ritual that it is `` less likely that we shall indulge them through our own acts ''.

And as they go through college, the students tend to bring their political position in line with that prevalent in the social groups to which they belong.

Scribe a line through the center of the pin and across the face of tappet 3, parallel to piece `` A ''.

The roots of this equation are just the ordinates of the intersections of the graph of B with a straight line of unit slope through Af in the b-plane ( the plane of the graph of b ).

A general line L meets Q in two points, Af and Af, through each of which passes a unique generator of the regulus, Af, whose lines are simple secants of Aj.

Hence, thought of as a line in a particular plane **yp, any tangent to Q has a unique image and moreover this image is the same for all planes through L.

A line through two of these points, Af and Af, will be transformed into the entire bilinear congruence having the tangents to **zg at Af and Af as directrices.

Since a ruled surface of order N with N concurrent generators is necessarily a cone, it follows finally that every line through a point, P, of **zg meets its image at P, as asserted.

The first possibility requires that there be a line through P which meets **zg in Af points ; ;

the second requires that there be a line through P which meets **zg in Af points.

If **ya is the multiple secant of **zg which passes through Af and **yb is the simple secant of **zg which passes through Af, and if Af are the points in which **ya meets **zg, and if Af is the image of Af on the generator **yb, it follows that the image of the line Af is Af.

Word reached the company that the man behind these depredations was Manuel Gonzales, a man with many followers, including a number who were kept in line through fear of him.

If the cutting force, Af, is plotted against film thickness, a straight line should result passing through the origin and having slope Af.

Often, threading through the overcast, he was forced to fly close to the ground by a low ceiling, skimming above the Winooski or the White River along the line of the broken railroad.

Saxton has made only one second-half appearance this season and that was in the Washington State game, for four plays: he returned the kickoff 30 yards, gained five yards through the line and then uncorked a 56-yard touchdown run before retiring to the bench.

The division said it would be impossible to work on the line until then because of the large amount of acid sewage from jewelry plants in the area flowing through the line, heavy vehicle traffic on Eddy Street and tide conditions.

Once four Tibetan monks, in their saffron robes, filed through the cafeteria line.

It was like a long thin line drawn through a pink circle.

In this line, Mohamed Temam, Abdelkhader Houamel have also returned through this art, scenes from the history of the country, the habits and customs of the past and the country life.

A line running from Cape Aspro ( 28 ° 16 ' E ) in Asia Minor, to Cum Burnù ( Capo della Sabbia ) the Northeast extreme of the Island of Rhodes, through the island to Cape Prasonisi, the Southwest point thereof, on to Vrontos Point ( 35 ° 33 ' N ) in Skarpanto < nowiki >< nowiki ></ nowiki >, through this island to Castello Point, the South extreme thereof, across to Cape Plaka ( East extremity of Crete ), through Crete to Agria Grabusa, the Northwest extreme thereof, thence to Cape Apolitares in Antikithera Island, through the island to Psira Rock ( off the Northwest point ) and across to Cape Trakhili in Kithera Island, through Kithera to the Northwest point ( Cape Karavugia ) and thence to Cape Santa Maria () in the Morea.

These are defined by a simple involutorial transformation of the points in which a general line meets a nonsingular quadric surface bearing a curve of symbol Af.

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.

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.

A conic, C, being a ( 1, 1 ) curve on Q, meets the image of any line of Af, which we have already found to be a Af curve on Q, in Af points.

The preceding observations make it clear that there exist line involutions of all orders greater than 1 with no complex of invariant lines and with a complex of singular lines consisting exclusively of the lines which meet a twisted curve Aj.

The single curve line represents a specific formulation in a test example.

In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.

Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.

In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.

" The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.

The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem counter to everyday experience.

The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width.

But these are physical representations of the corresponding mathematical entities ; the line and the curve are idealized concepts whose width is 0 ( see Line ).

Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞.

The simplest method for scan converting ( rasterizing ) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments.

A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line segment to within a small tolerance.

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

* The curve is a straight line if and only if all the control points are collinear.

The corresponding form of the fundamental theorem of calculus is Stokes ' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

* Chord ( geometry ), a line segment joining two points on a curve