Damian located the enemy ahead of them, shooting intersections clear as they reached them.
He started the car again and drove through a series of tunnels and intersections, a virtual underground street grid, before arriving at a large garage filled with gleaming cars.
The robed man led her into the fortress and wound his way through bright intersections, down stairs, and into a more opulent part of the building.
They crossed more intersections, descended to the level below, and stopped outside of double doors.
Several more warriors stood at intersections like gargoyles, moving only to point in the direction she needed to go.
Accompanied by two guards, she mounted her favorite bay horse and pounded through familiar roads and intersections to the southern wall., The chill of the ocean crept into its walls.
The image of the star is set updn the intersections of the lines of the central cross, and the positions of the reseau-lines are read off by estimation to - of a division on the glass scale.
These intersections determine the centres of the semicircles CC which form the ends of the respective knuckles.
Cubic equations were solved geometrically by determining the intersections of conic sections.
In these instruments the lines are ruled upon a spherical surface of speculum metal, and mark the intersections of the surface by a system of parallel and equidistant planes, o; of which the middle member passes through the centre of the sphere.
The locus of these intersections is the quadratrix.
Archytas of Tarentum (c. 430 B.C.) solved the problems by means of sections of a half cylinder; according to Eutocius, Menaechmus solved them by means of the intersections of conic sections; and Eudoxus also gave a solution.
This result is modified if the action of the load near the section is distributed to the bracing intersections by rail and cross girders.
But if the load is distributed to the bracing intersections by rail and cross girders, then the shear at C' will be greatest when the load extends to N, and will have the values wXADN and -wXNEB.
The first pair of intersections may be either real or imaginary; we proceed to discuss the second pair.
A further deduction from the principle of continuity follows by considering the intersections of concentric circles.
- The " centres of similitude " of two circles may be defined as the intersections of the common tangents to the two circles, the direct common tangents giving rise to the " external centre," the transverse tangents to the " internal centre."
Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as radii, are members of the coaxal system.
The most brilliant are situated at the intersections of the inner halo and the parhelic circle; these are known as parhelia (denoted by the letter p in the figures) (from the Gr.
In this way he established the famous theorem that the intersections of the three pairs of opposite sides of a hexagon inscribed in a conic are collinear.
Corners of signs and intersections of lines were first fixed by minute tube-drill holes, into which the hand tool butted, so that it should not slip over the outer surface.
Descartes used the curve to solve sextic equations by determining its intersections with a circle; mechanical constructions were given by Descartes (Geometry, lib.
John Wallis utilized the intersections of this curve with a right line to solve cubic equations, and Edmund Halley solved sextic equations with the aid of a circle.
36 these various lines and planes are represented by their intersections with a uiiit sphere having 0 as centre.
Now suppose that a body receives first a positive rotation a about OA, and secondly a positive rotation e3 about OB; and let A, B be the intersections of these axes with a sphere described about 0 as centre.
80 by means of the intersections with a concentric spherical surface.
If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their " perpendicular height of incidence," i.e.
Ancient, but now extinct, volcanic upheavals are pretty common at the intersections of the main range with the transverse ranges; of these the most noteworthy are Elbruz and Kasbek.
Cramer, that, when a certain number of the intersections of two algebraical curves are given, the rest are thereby determined.
Gergonne had shown that when a number of the intersections of two curves of the (p+q)th degree lie on a curve of the pth degree the rest lie on a curve of the qth degree.
High, consisting of a six-sided lantern and royal crown, both sculptured, and resting on the intersections of two arched ornamental slips rising from the four corners of the top of the tower.
The intersections of two curves are obtained by combining their equations; viz.
The elimination from the two equations of y (or x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant equation in x (or y).
The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry - and it in fact plays an all-important part in modern geometry.
The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m= 2, if the hyperbola xy - 1= o be cut by the line y=0, the resultant equation in x is Ox- 1 = o, and there is apparently only the intersection (x 110, y =0); but the theorem is, in fact, true for every line whatever: a curve of the order in meets every line whatever in precisely m points.
We have, in the case just referred to, to take account of a point at infinity on the line y=0; the two intersections are the point (x=110, y=0), and the point at infinity on the line y= 0.
Intersections of Circles.
The construction in fact is, join the two points in which the third circle meets the first arc, and join also the two points in which the third circle meets the second arc, and from the point of intersection of the two joining lines, let fall a perpendicular on the line joining the centre of the two circles; this perpendicular (considered as an indefinite line) is what Gaultier terms the " radical axis of the two circles "; it is a line determined by a real construction and itself always real; and by what precedes it is the line joining two (real or imaginary, as the case may be) intersections of the given circles.
The intersections which lie on the radical axis are two out of the four intersections of the two circles.
The question as to the remaining two intersections did not present itself to Gaultier, but it is answered in Jean Victor Poncelet's Traite des propeietes projectives (1822), where we find (p. 49) the statement, "deux circles places arbitrairement sur un plan ...
We find in it explicitly the two correlative definitions: " a plane curve is said to be of the p ith degree (order) when it has with a line m real or ideal intersections," and " a plane curve is said to be of the mth class when from any point of its plane there can be drawn to it m real or ideal tangents."
The points of contact are found as the intersections of the curve u= o by a curve depending on the position of the arbitrary point, and called the " first polar " of this point; the order of the first polar is = m - r, and the number of intersections is thus =m(m - I).