Angle Sentence Examples

From this angle, it's hard to tell you from the mules.

From that angle, she could see the place on the mountain.

Brady muttered a curse, reached for his laser weapon, and fired at an angle at the ceiling.

Now, madam, these triangles are equal; please note that the angle ABC...

If you angle your sword, the blow glances off.

The point of the angle was broken off.

The angle between the rivers was now almost full.

He must have thought it fruitless to pursue that angle so he moved on.

Above them, the headlights of the two vehicles grew more distant, finally hidden from view by their angle of descent.

An elevation of great extent which rises at a very gentle angle from a surrounding depression is termed a " rise," one which is relatively narrow and steep-sided a " ridge," and one which is approximately equal in length and breadth but steep-sided a " plateau," whether it springs direct from a depression or from a rise.

The angle between the costa and termen is the apex.

The same process is then repeated in another direction, so that the new bands cross the old at an angle adapted to the nature of the design.

The under jaw, from the chin about as far back as the angle of the mouth, is generally white.

Join CR; the angle CRQ is then the eccentric anomaly.

The railway up the right bank of the Nile was continued to Kerma, in order to evade the difficulties of the 3rd cataract; but the sirdar had conceived the bold project of cutting off the great angle of the Nile from Wadi Haifa to Abu Hamed, involving nearly 600 m.

Wollaston in the angles of the rhombohedra of the carbonates isomorphous with calc-spar, he observed that the angle in the case of calc-spar varied with the temperature.

The cylinders are made of steel plate, lined with refractory bricks, are carried on rollers at a slight angle with the horizontal, and are rotated by power.

Technically that's true, but maybe she has an angle.

He looked for signs anyone had walked through the brush and branches, pausing at intervals to squat and look at his surroundings from a different angle.

Assuming the four valencies of the carbon atom to be directed from the centre of a regular tetrahedron towards its four corners, the angle at which they meet.

Baeyer supposes that in the formation of carbon, rings " the valencies become deflected from their positions, and that the tension thus introduced may be deduced from a comparison of this angle with the angles at which the strained valencies would meet.

The speed of the ship must therefore be so regulated that the angle of immersion is as great as the inclination of the steepest slope passed over.

Besides these offshoots of the Apennines there are in this part of Central Italy several detached mountains, rising almost like islands on the seashore, of which the two most remarkable are the Monte Argentaro on the coast of Tuscany near Orbetello (2087 ft.) and the Monte Circello (1771 ft.) at the angle of the Pontine Marshes, by the whole breadth of which it is separated from the Volscian Apennines.

But some stems grow parallel to the surface of the soil, while the branches both of stems and roots tend to grow at a definite angle to the main axis from which they come.

The angle which the earth's axis makes with the plane in which the planet revolves round the sun determines the varying seasonal distribution of solar radiation over the surface and the mathematical zones of climate.

This arises mostly from the angle formed by the keel with the body of the sternum, passes by a strong tendon through the foramen triosseum, and is inserted upon the upper tubercle of the humeral crest, which it rotates and abducts.

The article REFLECTION explains the symmetrical arrangement of images formed by two mirrors inclined at an angle which is a sub-multiple of four right angles.

In this case the rack had pin teeth carried in a pair of angle bars.

In Great Britain the curvature is defined by stating the length of the radius, expressed in chains (i chain=66 ft.), in America by stating the angle subtended by a chord ioo ft.

Another of Yung-lo's bells is hung in a Buddhist temple outside the north-west angle of the city wall, and is covered both on the inside and outside with the Chinese texts of the Lankavatara Sutra, and the Saddharma pundarika Sutra.

But he had, as Newman afterwards said of him, "struck into the movement at an angle."

A distinctive feature is the position assumed in resting; Culex has a humpbacked attitude, while in Anopheles the proboscis, head and body are in a straight line, and in many species inclined at an angle to the wall, the tail sticking outwards.

Among the most important of these were - Lyttus or Lyctus, in the interior, south-east of Cnossus; Rhaucus, between Cnossus and Gortyna; Phaestus, in the plain of Messara, between Gortyna and the sea; Polyrrhenia, near the north-west angle of the island; Aptera, a few miles inland from the Bay of Suda; Eleutherna and Axus, on the northern slopes of Mount Ida; and Lappa, between the White Mountains and the sea.

The two great rivers of China, the Hwang-ho and the Yang-tsze-kiang take their rise from the eastern face of Tibet, the former from the north-east angle, the latter from the south-east.

The southern and south-western face follows the coast closely up the Persian Gulf from the mouth of the Indus, and is formed farther west by the mountain scarp, which, rising in many points to 10,000 ft., flanks the Tigris and the Mesopotamian plains, and extends along Kurdistan and Armenia nearly to the 40th meridian; beyond which it turns along the Taurus range, and the north - eastern angle of the Mediterranean.

These double and triple ranges, which have a general elevation of 8500-10,000 ft., stretch from the south-east angle of the Black Sea, 400 m.

The anomaly is then the angle BFP which the radius vector makes with the major axis.

The anomaly AFQ of Q at any moment is called the mean anomaly, and the angle QFP by which the true anomaly exceeds it at that moment is the equation of the centre.

One of these is the position of the line MN through the sun at F in which the plane of the orbit cuts some fundamental plane of reference, commonly the ecliptic. This is called the line of nodes, and its position is specified by the angle which it makes with some fixed line FX in the fundamental plane.

The angle from the pericentre to the actual radius vector, and the length of the latter being found, the angular distance of the planet from the node in the plane of the orbit is found by adding to the true anomaly the distance from the node to the pericentre.

If a small magnet of moment M is placed in the sensibly uniform field H due to a distant magnet, the couple tending to turn the small magnet upon an axis at right angles to the magnet and to the force is MH sin 0, (17) where 0 is the angle between the axis of the magnet and the direction of the force.

The potential at any point due to a magnetic shell is the product of its strength into the solid angle w subtended by its edge at the given point, or V = Fu.

M/H = (d 2 -1 2) tan 0/2d, where 1 is half the length of the magnet, which is placed in the " broadside-on " position as regards a small suspended magnetic needle, d the distance between the centre of the magnet and the needle, and 0 the angle through which the needle is deflected by the magnet.

Thus if the magnet is suspended horizontally by a fine wire, which, when the magnetic axis points north and south, is free from torsion, and if 0 is the angle through which the upper end of the wire must be twisted to make the magnet point east and west, then MH = CB, or M = C6/H, where C is the torsional couple for r 0.

Let 0 be the angle which the standard magnet M makes with the meridian, then M'/R = sin 0, and M/R = cos 0, whence M' = M tan 0.

The direction of the resultant field of force will then make, with that of H E, an angle 0, such that Hp/H E tan 0, and the suspended needle will be deflected through the same angle.

The angle B is indicated by the position of the spot of light upon the scale, and the horizontal intensity of the earth's field H E is known; thus we can at once determine the value of H P, from which the magnetization I of the body under test may be calculated.

The effect of the ends of the wire is, as Ewing remarks, to shear the diagram in the horizontal direction through the angle which the sloping line makes with the vertical.

The curve thus constructed should be a straight line inclined to the horizontal axis at an angle 0, the tangent of which is 1.6.

If a coil of insulated wire is suspended so that it is in stable equilibrium when its plane is parallel to the direction of a magnetic field, the transmission of a known electric current through the coil will cause it to be deflected through an angle which is a function of the field intensity.

Maxwell (Electricity and Magnetism, § 444), recognizing that the theory in this form gave no account of residual magnetization, made the further assumption that if the deflection of the axis of the molecule exceeded a certain angle, the axis would not return to its original position when the deflecting force was removed, but would retain a permanent set.

The ovary is generally two-chambered, with two inverted ovules standing side by side at the inner angle of each chamber.

His investigation of the properties of amicable numbers and of the problem of trisecting an angle, are of importance.

The Treaty of Trianon satisfied the most essential claims of Yugoslavia, by dividing the whole Banat (save a small Magyar triangle opposite the city of Szeged) between her and Rumania, and by assigning to her the whole Backa (except Baja and district), part of the Baranya (forming the angle between Drave and Danube) and the Medjumurje (between Drava and Mur).

We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture.

If the angle subtended by the components of a double line be twice that subtended by the wave-length at a distance equal to the horizontal aperture, the central bands are just clear of one another, and there is a line of absolute blackness in the middle of the combined images.

From a measurement of the maximum distance the least angle between consecutive lines consistent with resolution may be deduced, and a comparison made with the rule stated above.

The extreme value possible for a is a right angle, so that for the microscopic limit we have Z X o/µ (2).

If 2R be the diameter of the objectglass and D the distance of the object, the angle subtended by AP is E/D, and the angular resolving power is given by X/2 D sin a = X/2 R (3) This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.

The eye, unaided or armed with a telescope, is able to see, as points of light, stars subtending no sentsible angle.

B) - f S,u ds (along The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance AB.

In another direction, making a small angle with the last, such that the projection of AB upon it amounts to a few wavelengths, it is easy to see that the mode of interference is the same as if there were no ruling.

The sine of an angle can never be greater than unity; and consequently under the most favourable circumstances only 1/m 2 ir 2 of the original light can be obtained in the m u ' spectrum.

Hence, if a be the width of the diffracted beam, and do the angle through which the wave-front is turned, ado = dX, or dispersion = /a ..

For a disturbing force of given integral magnitude it is seen to be everywhere about an axis perpendicular to r and the direction of the force, and in magnitude dependent only upon the angle (43) between these two directions and upon the distance (r).

The proportionality of the secondary disturbance to sin 43 is common to the present law and to that given by Stokes, but here there is no dependence upon the angle 0 between the primary and secondary rays.

If the ray suffers one internal reflection at D, then it is readily seen that, if DB be the path of the reflected ray, the angle ADB equals 2r, i.e.

At the Louvre, Goujon, under the direction of Lescot, executed the carvings of the south-west angle of the court, the reliefs of the Escalier Henri II., and the Tribune des Cariatides, for which he received 737 livres on the 5th of September 1550.

The government offices are principally in Whitehall, the fine thoroughfare which connects Parliament Square, in the angle between the Houses and the Abbey, with Trafalgar Square.

Besides the forum Stukeley suggested the sites of seven other buildings - the Arx Palatina guarding the south-eastern angle of the city where the Tower now stands, the grove and temple of Diana on the site of St Paul's, &c. No traces of any of these buildings have been found, and they are therefore purely conjectural.

Then, as the chronicler writes, " all the Angle race turned to him (Alfred) that were not in bondage of the Danish men."

The ore is mined in the ordinary way, by pick and shovel if soft, or by the aid of powder if necessary, and the funnel-shaped bottom of the pit is maintained at such an angle that little or no shovelling is required to bring the excavated material to the shaft.

The stability of a ship is investigated practically by inclining it; a weight is moved across the deck and the angle is observed of the heel produced.

For if the ship turns through a small angle 0 about the line FF', then b1, b 2, the C.G.

The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF', at a distance FK from F FK= (k2-hV/A)/FQ sin QFF' (2) through an angle 0 or a slope of one in m, given by P sin B= m wA FK - W'Ak 2V hV FQ sin QFF', (3) where k denotes the radius of gyration about FF' of the water-line area.

I n a straight uniform current of fluid of density p, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle 0 with the velocity, is oAq cos 0, the product of the density p, the area A, and q cos 0 the component velocity normal to the plane.

Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and 0 the angle which the outward-drawn normal makes with the velocity q at that point, dM/dt = rate of increase of fluid inside the surface, (I) =flux across the surface into the interior _ - f f pq cos OdS, the integral equation of continuity.

Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = o, = o; and in steady motion the equations reduce to dH/dv=2q-2wn = 2gco sin e, (4) where B is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dv is drawn perpendicular.

Thus if d,/ is the increase of 4, due to a displacement from P to P', and k is the component of velocity normal to PP', the flow across PP' is d4 = k.PP'; and taking PP' parallel to Ox, d,, = vdx; and similarly d/ ' = -udy with PP' parallel to Oy; and generally d4,/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.

The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.

If the direction of motion makes an angle 0' with Ox, tan B' = d0 !dam _ ?xy 2 = tan 20, 0 =-10', (9) dy/ y and the velocity is Ua2/r2.

Denoting the cross-section a of a filament by dS and its mass by dm, the quantity wdS/dm is called the vorticity; this is the same at all points of a filament, and it does not change during the motion; and the vorticity is given by w cos edS/dm, if dS is the oblique section of which the normal makes an angle e with the filament, while the aggregate vorticity of a mass M inside a surface S is M - l fw cos edS.

The method of electrical images will enable the stream function, )' to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle 7r/m, where m is an integer (R.

Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving S the Angle of Rifling, and n the Pitch of Rifling in Calibres.

The curve also permits the solution of the problems of duplicating a cube and trisecting an angle.

When crystallized from water, crystals belonging to the orthorhombic system, and having a prism angle of 61 0 10', are obtained; they are often twinned on the prism planes, giving rise to pseudo-hexagonal groups resembling aragonite.

Passau consists of the town proper, lying on the rocky tongue of land between the Danube and the Inn, and of four suburbs, Innstadt on the right bank of the Inn, Iizstadt on the left bank of the Ilz, Anger in the angle between Ilz and the Danube, and St Nikola.

The Anophelinae have narrow bodies, and generally spotted wings, and when at rest keep body and proboscis in a straight line, often at a considerable angle with the supporting surface; in this way they can be distinguished from Culicinae, which have a humped-up thorax with which the proboscis forms an angle, and in the resting position keep the body parallel to the support.

Trieste is situated at the northeast angle of the Adriatic Sea, on the Gulf of Trieste, and is picturesquely built on terraces at the foot of the Karst hills.

It owes its development to its geographical situation in the north-east angle of the Adriatic Sea at the end of the deeply indented gulf, and to its harbour, which was more accessible to large vessels than that of Venice.

The county is well watered by numerous streams. The principal are the Callan, the Tynan and the Tallwater, flowing into the Blackwater, which, after forming the boundary between this county and Tyrone, empties itself into the south-western angle of Lough Neagh.

The upper angle of it consists of rich and fertile districts, such as Murshidabad, Nadia, Jessore and the 24 Parganas.

Hence the total solid angle round any point is 47r.

The solid angles subtended by all normal sections of a cone at the vertex are therefore equal, and since the attractions of these sections on a particle at the vertex are proportional to their distances from the vertex, they are numerically equal to one another and to the solid angle of the cone.

The electric density on the sphere being uniform, the quantities of electricity on these areas are proportional to the areas, and if the electric force varies inversely as the square of the distance, the forces exerted by these two surface charges at the point in question are proportional to the solid angle of the little cone.

At each point on this surface the resultant force has a certain value, and a certain direction inclined at an angle 0 to the normal to the selected surface at that point.

The normal section of the cone at that point is equal to dS cosO, and the solid angle dw is equal to dS cos0/x 2.

Accordingly, since the total solid angle round a point is 47r, it follows that the total flux through the closed surface due to the single point charge q is 41rq, and what is true for one point charge is true for any collection forming a total charge Q of any form.

On the top of this system as a foundation were set several statue bases, one bearing the signature of Lysippus, which shows that the system stood there at least as early as the 4th century B.C. Some parts of it may have been taken from older buildings, but not the cornice nor the corner metope block which formed an obtuse angle.

The city possesses five gates, two on the northern face, the Kutab-chak near the north-east angle of the wall, and the Malik at the re-entering angle of the Ark-i-nao; and three others in the centres of the remaining faces, the Irak gate on the west, the Kandahar gate on the south and the Kushk gate on the east face.

Chaumont is picturesquely situated on an eminence between the rivers Marne and Suize in the angle formed by their confluence.

He found out the formula for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sines.

His war-ships were arranged in what was called a half-moon, and was in fact an obtuse angle with his flagship, the "Brederode," at the apex.

From a series of measures of the angle between Jupiter's satellites and the planet, made in June and July 1794 and in August and September 1795, Schur finds the mass of Jupiter =I / Io 4 8.55 1.45, a result which accords well within the limits of its probable error with the received value of the mass derived from modern researches.

In 1812 Bessel measured with it the angle between the components of the double star 61 Cygni and observed the great comet of 1811.

This construction would give all the advantage of the younger Dollond's object-glass micrometer, and more than its sharpness of definition, without liability to the systematic errors which may be due to want of homogeneity of the object-glass; for the lenses will not be turned with respect to each other, but, in measurement, will always have the same relation in position angle to the line joining the objects under observation.

This ring is necessary in order to allow the rods to follow the micrometer heads when the position angle is changed.

For this purpose the position angle of the eye-piece micrometer is set to that of the head, and the eye-piece is displaced from the axis of the tube (in the direction of the movable segment) by an amount equal to half the angle under measurement.

The eye-piece is fixed in the axis, and the segments are symmetrically displaced from the axis each by an amount equal to half the angle measured.

The chief objections to the method are that, as one star is in the axis of the telescope and the other displaced from it, the images are not both in focus of the eye-piece,3 and the rays from the two stars do not make the same angle with the optical axis of each segment.

Their slow motion in position angle, however, was not all that could be desired.

The circles for position angle and declination are read by micrometer-microscopes illuminated by the lamp L; the scales are illuminated by the lamp 1.

The tube V, on the contrary, is attached to the cradle, and merely forms a support for the finder Q, the handles at f and p, and the moving ring P. The latter gives quick motion in position angle; the handles at p clamp and give slow motion in position angle, those at f clamp and give slow motion in right ascension and declination.

To remedy drawback (2) Repsolds provided for the Yale heliometer an additional handle for motion in position angle, intermediate in velocity between the original quick and slow motions.

With similar bevel-gear and rods the tangent screw is connected to the hand-wheel, 79, by which the observer communicates the fourth or slowest motion in position angle.

In observations of position angle one of the two tablets 81 can be used to record the readings.

Having selected the most suitable one he directs the axis of the finder to the estimated middle point between the comet and the star, turns the finder-micrometer in position angle until the images of comet and star lie symmetrically between the parallel position wires, and then turns the micrometer screw (which moves the distance-wires symmetrically from the centre in opposite directions) till one wire bisects the comet and the other the star.

Simi larly the prism may be used for the study and elim- " ination of personal errors depending on the angle made s by a double star with the vertical.

Dawes found the best method for the purpose in question was to limit the aperture of the object-glass by a diaphragm having a double circular aperture, placing the line joining the centres of the circles approximately in the position angle under measurement.

By a scale attached to the sliding tube the magnifying power of the eye-piece was deduced, and this combined with the angle of the prism employed gave the angle measured.

If p" is the refracting angle of the prism, and n the magnifying power of the eye-piece, then p"ln will be the distance observed.

In the same way a mean angle of slope of approximately half a degree was found for the Adriatic and the Black Sea.

Where the French telegraph cable between Brest and New York passes from the continental shelf of the Bay of Biscay to the depths of the Atlantic the angle of slope is.

The Australian Shelf rises steeply as a rule from depths, of 2500 to 3000 fathoms. A broad depression with depths of from 3300 to 3500 fathoms lies to the east of the Cocos Islands and extends into the angle between the Malay Archipelago and Australia.

Mill has shown that in the North Sea off the Firth of Forth the average depth of visibility of a disk in the winter half-year was 4; fathoms and in the summer half-year 62 fathoms, and, although the greater frequency of rough weather in winter might tend to obscure the effect, individual observations made it plain that the angle of the sun was the main factor in increasing the depth to which the disk remained visible.

The use of a sliding weight is not recommended in depths much exceeding 200 fathoms on account of the time required and the risk of the line sagging at a low angle and so stopping the weight.

The inner face is smooth, but the back is strengthened with angle brackets at the corners.

In the working of thick seams inclined at a high angle, such as those in the south of France, and in the lignite mines of Styria and Bohemia, the method of working in horizontal slices, about i 2 or r 5 ft.

The ratio p is given by e"` e, where e= 2.718; µ is the coefficient of friction and 0 the angle, measured in radians,, subtended by the arc of contact between the rope and the wheel.

If the angle 0 1 =0 2 =120 0, Q = (P - p) neglecting friction.

D is the dispart sight, S the tangent sight, A'DS the clearance angle.

Guns without dispart sights cannot be layed at elevations below the clearance angle.

Now the height to which the tangent sight has been raised in order to direct the axis on G' is evidently proportional to the tangent of the angle OMS =AXS.

This angle is called the angle of elevation; OM is constant and is FIG.

The formula for length of scale is, length = sighting radius X tangent of the angle of elevation.

If now the telescope be directed on the target and this level be brought to the centre of its run, the angle of sight can be read - if afterwards any range ordered is put on the sight and the gun truly layed, this bubble will be found in the centre of its run - so that if thereafter the target becomes obscured the gun can be relayed by elevating till the bubble is in the centre of its run, or at a completely concealed target the angle of sight can, if the range and difference of level are known or can be measured from somewhere near the gun, be put on by means of the micrometer screw, and the gun subsequently layed by putting the range in yards or degrees on the sight drum and elevating or depressing till the bubble is central.

Range is reduced by increasing the angle of elevation (by clinometer) or by using reduced charges.

In coast defence artillery, owing to the fact that the guns are on fixed mountings at a constant height (except for rise and fall of tide) above the horizontal plane on which their targets move, and that consequently the angle of sight and quadrant elevation for every range can be calculated, developments in sights, in a measure, gave way to improved means of giving quadrant elevation.

There must be two sets of elevating gears, one which brings the axis of the gun and the sights together on to the target, thus finding the angle of sight and also pointing the axis of the gun at the target, and a second by which, independent of the sight which remains fixed, the elevation due to the range can be given to the gun and read by means of a pointer and dial marked in yards for range.

Then ABF = angle of elevation; EFB =quadrant angle; BCD=angle of sight; EBF=ABF - ABE; and since ABE = BCD, it also equals ABF - BCD.

The sights themselves fit into sockets cut at the proper angle for drift, and are raised in their sockets the requisite amount for the range by means of a small hand-wheel; they are thus non-recoiling sights.

In Hippotragus the stout and thickly ringed horns rise vertically from a ridge above the eyes at an obtuse angle to the plane of the lower part of the face, and then sweep backwards in a bold curve; while there are tufts of long white hairs near the eyes.

If a .JP solid circle be fixed in any one position and a tube be pivoted on its centre so as to move; and if the line C D be drawn upon the circle pointing towards any object Q in the heavens which lies in the plane of the circle, by turn ing the tube A B towards any other object P in the plane of the circle, the angle B 0 D will be the angle subtended by the two objects P and Q at the eye.

The conception of the right angle is strengthened, by contrast, by the use of bricks in the form of a rhombus.

The treatment of an angle as generated by rotation, the investigation of the relations between trigonometrical ratios and circular measure, the application of interpolation to trigonometrical tables, and the general use of graphical methods to represent continuous variation, all imply an analytical onlook, and must therefore be deferred to this stage.

The revolution need not be complete, but may be through any angle.

Let a be the radius of a circle, and 0 (circular measure) the unknown angle subtended by an arc. Then, if we divide 0 into m equal parts, and L 1 denotes the sum of the corresponding chords, so that L i =2ma sin (0/2m), the true length of the arc is L1 +a9 3 - 5 + ..., where cp. =B/2m.

The tangent of the angle of deflection 0 of this needle measured from its position, when the shunt coil is disconnected, is equal to the ratio of the voltage of the dynamo to the current through the insulator.

In the other gap are pivoted two coils wound on an iron core and connected at nearly a right angle to each other.

Kb(30s, a cube), in geometry, a solid bounded by six equal squares, so placed that the angle between any pair of adjacent faces is a right angle.

It became known as the "Delian problem" or the "problem of the duplication of the cube," and ranks in historical importance with the problems of "trisecting an angle" and "squaring the circle."

Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the language of algebraic geometry, is to solve geometrically a cubic equation.

According to Proclus, a man named Hippias, probably Hippias of Elis (c. 460 B.C.), trisected an angle with a mechanical curve, named the quadratrix.

A small stream called the Crkvina enters the Vrbas from the north-east and in the angle thus formed stand the citadel and barracks, with the 16th-century Ferhadiya Jamia, largest and most beautiful of more than 40 mosques in the city.

A ray making an angle less than 0 with the tangent will, with its reflections, touch a larger circle.

It further follows, as in the analogous case of light, that there is a certain angle termed the critical angle, whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflection back into the first body, if the.

If 0 is the angle of twist, the angular velocity is d0/dt.

The west and south-west half of Saxony is more or less occupied by the ramifications and subsidiary groups of this range, one of which is known from its position as the Central Saxon chain, and another lower group still farther north as the Oschatz group. The south-east angle of Saxony is occupied by the mountains of Upper Lusatia (highest summit 2600 ft.), which form the link between the Erzgebirge and Riesengebirge in the great Sudetic chain.

If r, r i be the radii of two spheres, d the distance between the centres, and 0 the angle at which they intersect, then d2=r2+ r12 2rr l cos ¢ hence 2rr 1 cos =d2r2 - r22.

It consists of a pair of tubular girders with solid or plate sides stiffened by angle irons, one line of rails passing through each tube.

The anchors are built up of steel plates and angle bars, and are buried in a large mass of concrete.

Soft steel is used for rivets always, and sometimes for the whole superstructure of a bridge, but medium steel more generally for the plates, angle bars, &c., the weight of the bridge being then reduced by about 7% for a given factor of safety.

R=wx (I+x2/4Y2) Let i be the angle between the tangent at any point having the co-ordinates x and y measured from the vertex, then 3..

The county is roughly triangular in form, London lying at the apex of the western angle, the North Foreland at that of the eastern and Dungeness at that of the southern.

In 1845 began the marked influx of Germans, which lasted in large degree up to 1860; they first limited themselves to the district "Over the Rhine" (the Rhine being the Miami & Erie Canal), in the angle north-east of the junction of Canal and Sycamore streets, but gradually spread throughout the city, although this "Over the Rhine" is still most typically German.

The factor T becomes of importance in long range high angle fire, where the shot reaches the higher attenuated strata of the atmosphere; on the other hand, we must take about 800 in a calculation of shooting under water.

The last function A, called the altitude function, will be explained when high angle fire is considered.

So also the angle /3 must be increased by S to obtain the angle at which the shot strikes a horizontal plane - the water, for instance.

It is found that the alteration of the tangent elevation is almost insensible, but the quadrant elevation requires the addition or subtraction of the angle of sight.

Replacing then the angle i on the right-hand side of equations (54) - (56) by some mean value, t, we introduce Siacci's pseudovelocity u defined by (59) u = q sec, t, so that u is a quasi-component parallel to the mean direction of the tangent, say the direction of the chord of the arc.

The equations (66) - (71) are Siacci's, slightly modified by General Mayevski; and now in the numerical applications to high angle fire we can still employ the ballistic table for direct fire.

Siacci's altitude-function is useful in direct fire, for giving immediately the angle of elevation 4, required for a given range of R yds.

After a certain discount for friction and the recoil of the gun, the net work realized by the powder-gas as the shot advances AM is represented by the area Acpm, and this is equated to the kinetic energy e of the shot, in foot-tons, (I) e d2 I + p, a in which the factor 4(k 2 /d 2)tan 2 S represents the fraction due to the rotation of the shot, of diameter d and axial radius of gyration k, and S represents the angle of the rifling; this factor may be ignored in the subsequent calculations as small, less than I %.

To the west of Parr's Ridge the surface for the most part slopes gently down to the east bank of the Monocacy river (which flows nearly at a right angle with the streams east of the Ridge), and then from the opposite bank rises rapidly toward the Catoctin Mountain; but just above the mouth of the Monocacy on the east side of the valley is Sugar Loaf Mountain, which makes a steep ascent of 1250 ft.

They required humouring, and their march discipline was very elastic. But in battle the "thinking bayonets" resolutely obeyed orders, even though it were to attack a Marye's Hill, or a "Bloody Angle," for they had undertaken their task and would carry it through unflinchingly.

The district of Ballia, constituted in 1879, occupies an angle at the junction of the Gogra with the Ganges, being bordered by two districts of Behar.

The triangle is here an irregular one, consisting of a narrow base to which one end of the string was fixed, while the second side, forming a slightly obtuse angle with the base, consisted of a wide and slightly curved sound-board pierced with holes through which the other end of the strings passed, being either knotted or wound round pegs.

The willows are cut at the first indication of the sap rising and "couched" in rotten peelings and soil at a slight angle, the butts being on the ground, which should be strewn with damp straw from a manure heap. The tops are covered lightly with rotted peelings and by periodical application of water, fermentation is induced at the bottom, heat is engendered, the leaves force their way through the covering and peeling may begin.

The angle of difference between these deviations is called the dispersion for those rays.

This quantity may readily be expressed in terms of the refractive indices for the three colours, for if A is the angle of the prism (supposed small) bc=(/1c - I)A, bD =(/ AD - OA, F - I)A, where µc, A n, µ F are the respective indices of refraction.

It is more useful than (1), as the refractive indices may be measured with a prism of any convenient angle.

If the increase of the angle of refraction were proportional to the diminution of wave-length for a prism of any material, the resulting spectrum would also be normal.

In studying the dispersion of the aniline dyes, a prism with a very small refracting angle is made of two glass plates slightly inclined to each other and enclosing a very thin wedge of the dye, which is either melted between the plates, or is in the form of a solution retained in position by surface-tension.

Only very thin layers are sufficiently transparent to show the dispersion near or within an absorption band, and a large refracting angle is not required, the dispersion usually being very considerable.

Those genera, like Paradoxides, Olenus, Asaphus, Phillipsia and others, in which this groove cuts the posterior edge of the head-shield on the inner side of its angle are referred to the Opisthoparia; those, like Dalmanites and Phacops, in which it cuts the lateral border in front of the posterior angle, belong to the Proparia.

But in certain genera, like Conocoryphe, Calymmene and Triarthrus, it cuts the margin of the head-shield so close to the posterior angle that the distinction between the two groups practically breaks down.

At one angle, a square pillared projection contains the marble fountain or monks' lavatory, evidently the work of Moslem sculptors.

The prior's group "entered at the south-east angle of the green court, placed near the most sacred part of the cathedral, as befitting the distinguished ecclesiastics or nobility who were assigned to him."

Extending from the south-east angle of the main group of buildings are the walls and foundations of a secondary group of considerable extent.

The earliest form of shutter weir, known as a bear-trap, introduced in the United States in 1818, and subsequently erected across the Marne in France, consists of two wooden gates, each turning on a horizontal axis laid across the apron, inclined towards one another and abutting together at an angle in the centre when the weir is closed; the up-stream one serves as the weir, and the down-stream one forms its support, and both fall flat upon the apron for opening the weir.'

The right and left hepatic ducts, while still in the transverse fissure, unite into a single duct which joins the cystic duct from the gall bladder at an acute angle.

Excavations have revealed one street and the north-west angle of the town walls, while the local museum contains over 2000 inscriptions, besides statues and other antiquities.

Mactridae.-External gill-plate directed ventrally; siphons united, invested by a chitinous sheath; foot long, bent at an angle, without byssus.

The base of the triangle is upward, and at each lateral angle one of the Fallopian tubes opens.

On making a mesial vertical section of the uterus the cavity is seen as a mere slit which is bent about its middle to form an angle the opening of which is forward.

In addition to this, its long axis forms a marked angle with that of the vagina, so that the whole uterus is bent forward or anteverted.

The round ligament is a cord of unstriped muscle which runs from the lateral angle of its own side of the uterus forward to the internal abdominal ring, and so through the inguinal canal to the upper part of the labium majus.

The second section, Castoroidea, of the present group includes only the family Castoridae, represented by the beavers, which are large aquatic rodents characterized by their massive skulls, devoid of post-orbital processes, with the angle of the lower jaw rounded, the molars rootless or semi-rooted, with re-entering enamel-folds, and one pair of premolars above and below.

The upper lip is cleft, the jugal lacks an inferior angle, the fore part of the skull is short and broad; the cheek-teeth are partially rooted, with external and internal enamel-folds, the soles of the feet are smooth, there are six pairs of teats, the clavicles are imperfect and the tail is not prehensile.

The jugal is without an inferior angle, and extends forwards to the lachrymal; the palate is contracted in front and deeply emarginate behind; the incisors are short, and the molars divided by continuous folds into transverse plates; and the two halves of the lower jaw are welded together in front.

Associated with this is the strength and sharpness of the lower jaw, the prominence and anterior pcsition of the masseteric ridge, and the depth of the ramus from the alveolar line to the angle.

If the bearer be stationary, rain-drops will traverse the tube without touching its sides; if, however, the person be walking, the tube must be inclined at an angle varying as his velocity in order that the rain may traverse the tube centrally.

Hence for the rain to centrally traverse the tube, this must be inclined at an angle BAD to the vertical; this angle is conveniently termed the aberration due to these two motions.

The mechanic's instrument known as a bevel consists of a rule with two arms so jointed as to be adjustable to any angle.

The measurement of the declination involves two separate observations, namely, the determination of (a) the magnetic meridian and (b) the geographical meridian, the angle between the two being the declination.

This arrangement is not very convenient, as it is difficult to protect the mirror from accidental displacement, so that the angle between the geometrical and magnetic axes may vary.

The method of measuring the horizontal component which is almost exclusively used, both in fixed observatories and in the field, consists in observing the period of a freely suspended magnet, and then obtaining the angle through which an auxiliary suspended magnet is deflected by the magnet used in the first part of the experiment.

What is known as the method of sines is used, for since the axes of the two magnets are always at right angles when the mirror magnet is in its zero position, the ratio M/H is proportional to the sine of the angle between the magnetic axis of the mirror magnet and the magnetic - = meridian.

The difference between the two sets of readings gives twice the angle which the magnetic axis of the mirror magnet makes with the magnetic meridian.

Omitting correction terms depending on the temperature and on the inductive effect of the earth's magnetism on the moment of the deflecting magnet, if 0 is the angle which the axis of the deflected magnet makes with the meridian when the centre of the deflecting magnet is at a distance r, then zM sin B=I+P+y2 &c., in which P and Q are constants depending on the dimensions and magnetic states of the two magnets.

This angle depends on the ratio of the magnetic moment of the needle b to the total force of the earth's field.

The direction assumed by the needle is not generally towards the geographical north, but diverges towards the east or west of it, making a horizontal angle with the true FIG.

Following the first chart of lines of equal variation compiled by Edmund Halley in 1700, charts of similar type have been published from time to time embodying recent observations and corrected for the secular change, thus providing seamen with values of the variation accurate to about 30' of arc. Possessing these data, it is easy to ascertain by observation the effects of the iron in a ship in disturbing the compass, and it will be found for the most part in every vessel that the needle is deflected from the magnetic meridian by a horizontal angle called the deviation of the compass; in some directions of the ship's head adding to the known variation of the place, in other directions subtracting from it.

Thus, when a degree on the card is observed opposite the lubber's point, the angle between the directior.

It is also provided with an azimuth circle or mirror and a shadow pin or style placed in the centre of the glass cover, by either of which the variable angle between the compass north and true north, called the "total error," or variation and deviation combined, can be observed.

According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was utilized by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

Following Euclid, a right angle is formed by a straight line standing upon another straight line so as to make the adjacent angles equal; any angle less than a right angle is termed an acute angle, and any angle greater than a right angle an obtuse angle.

A solid angle is definable as the space contained by three or more planes intersecting in a common point; it is familiarly represented by a corner.

The angle between two planes is termed dihedral, between three trihedral, between any number more than three polyhedral.

A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs on a sphere, and is measured by the angle between the planes containing the arcs and the centre of the sphere.

The angle between a line and a curve (mixed angle) or between two curves (curvilinear angle) is measured by the angle between the line and the tangent at the point of intersection, or between the tangents to both curves at their common point.

Galileo proceeded to measure the motion of a body on a smooth, fixed, inclined plane, and found that the law of constant acceleration along the line of slope of the plane still held, the acceleration decreasing in magnitude as the angle of inclination was reduced; and he inferred that a body, moving on a smooth horizontal plane, would move with uniform velocity in a straight line if the resistance of the air, and friction due to contact with the plane, could be eliminated.

The cloth is woven "one end up and two ends down," and as there are more picks of weft per inch than ends of warp the diagonal lines pass from selvage to selvage at an angle of less than 45 degrees.

Jeanette is the converse of jean, being a twill of "two ends up to one down"; the diagonal passes from selvage to selvage at a greater angle than 45 degrees and the warp makes the wearing surface.

At the south-east corner of the gymnasium, in the angle between the south and the east portico, was a Corinthian doorway, which a double row of columns divided into three passages.

A line drawn eastward behind the treasure-houses, from the Prytaneum at the north-west angle, would give about 275 yds.

The Exedra of Herodes Atticus stood at the north limit of the Altis, close to the north-east angle of the Heraeum, and immediately west of the westernmost treasure-house (that of Sicyon).

The Prytaneum was at the north-west corner of the Altis, in such a position that its south-east angle was close to the north-west angle of the Heraeum.

These images stood at the northern side of the Agora, in a row, which extended from the north-east angle of the Metroum to the gate of the private entrance from the Altis into the Stadium.

One, for the west part, passed from the south-west angle of the Heraeum to the south portico outside the south Altisjwall.

As the sun rises, the rays enter the prisms more and more obliquely, and the angle of minimum deviation increases; but since the emergent ray makes the same angle with the refracting edge as the incident ray, it follows that the parhelia will remain on the parhelic circle, while receding from the inner halo.

From the head of the Val Pellice the main chain runs north-west, and diminishes much in average height till it reaches the Mont Thabor (10,440 ft.), which forms the apex of a salient angle which the main chain here presents towards the west.

Hence the main watershed extends eastwards, culminating in the Aiguille de Scolette (11,500 ft.), but makes a great curve to the north-west and back to the south-east before rising in the Rochemelon (11,605 ft.), which may be considered as a re-entering angle in the great rampart by which Italy is guarded from its neighbours.

As regards the main divisions, three are generally distinguished; the Western Alps (chiefly French and Italian, with a small bit of the Swiss Valais) being held to extend from the Col de Tenda to the Simplon Pass, the Central Alps (all but wholly Swiss and Italian) thence to the Reschen Scheideck Pass, and the Eastern Alps (wholly Austrian and Italian, save the small Bavarian bit at the north-west angle) thence to the Radstadter Tauern route, with a bend outwards towards the south-east, as explained under (2) in order to include the higher summits of the SouthEastern Alps.

If the two hands be placed flat upon the table, in the angle between the two books, and the cloth pushed towards the corner, it will at once be rucked up into a fold which will follow a curve not unlike that of the Alps.

The east and west walls should run parallel to each other, and at right angles to that on the north side, in all the most favoured localities; but in colder or later ones, though parallel, they should be so far removed from a right angle as to get the sun by eleven o'clock.

But as the rays of light, even in passing through transparent glass, lose much of their energy, which is further weakened in proportion to the distance it has to travel, the nearer the plant can be placed to the glass the more perfectly will its functions be performed; hence the importance of constructing the roofs at such an angle as will admit the most light, especially sunlight, at the time it is most required.

The single-stemmed cordon may be trained horizontally, obliquely at any required angle, or vertically if required, the first two arrangements being preferable.

The half-fan is a combination of the two forms, but as regards pruning does not materially differ from the horizontal, as two opposite side branches are produced in succession upwards till the space is filled, only they are not taken out so abruptly, but are allowed to rise at an acute angle and then to curve into the horizontal line.

The Half fan mode of training, which is intermediate between horizontal and fan training, is most nearly allied to the former, but the branches leave the stem at an acute angle, a disposition supposed to favour the more equal distribution of the sap. Sometimes, as in fig.

Many remains of Roman buildings have been discovered within the modern town, and the amphitheatre is still visible in the southern angle.

They rest unconformably upon the Carboniferous beds, and have themselves been tilted to a steep angle.

The Norrevold Gade leads through the N6rretor y past the Folketeatre and the technical school to the Orsteds park, and from its southern end the Vestervold Gade continues through the Raadhus Plads, a centre of tramways, flanked by the modern Renaissance town hall (190,), ornamented with bronze figures, with a tower at the eastern angle.

When truly adjusted the theodolite measures the horizontal angle between any two objects, however much they may differ in altitude, as the pole star and any terrestrial object.

The Jumna and the Ganges enclose within their angle a fertile tract well irrigated with tanks and wells.

In the north-west angle of the walled enclosure stands Fort Santiago, which was built at the same time as the walls to defend the entrance to the river; the remaining space is occupied largely by a fine cathedral, churches, convents, schools, and government buildings.

Near the north-east angle of Intramuros is the Bridge of Spain, a stone structure across the Pasig, leading to Binondo, the principal shopping and financial district;.

The castle, at the north-east angle of the town, commands a fine view.

The upper angle formed between the leaf and the stem is called its axil; it is there that leaf-buds are normally developed.

The size of the angle between the median planes of two consecutive leaves in an alternate arrangement is their divergence; and it is expressed in fractions of the circumference of the axis which is supposed to be a circle.

If this arrangement is expressed by a fraction, the numerator of which indicates the number of turns, and the denominator the number of internodes in the spiral cycle, the fraction will be found to represent the angle of divergence of the consecutive leaves on the axis.

The angle of This is expressed by two-fifths.

This explains why it is so far from that building, which is situated at the south-eastern angle of the town, about 500 yds.

The phase angle, however, varies so much as to suggest that the term mainly represents local causes or observational uncertainties.

In the angle between the two rivers is the delta island of Antonio Vaz.

One of the simplest consists of a plane mirror rigidly connected with a revolving axis so that the angle be tween the normal to the mirror and the axis of the instrument equals half the sun's polar distance, the mirror being adjusted so that the normal has the same right ascension as the sun.

The axis of rotation AB bears a rigidly attached rod DBC inclined to it at an angle equal to the sun's polar distance.

It is equal to the angle at the pole between the hour circle through the body and the meridian, but is usually expressed in time.

The native principality of Kashmir occupies the north-western angle of India.

At the northeastern angle of that frontier, the Dihang river, the connecting link between the Tsanpo of Tibet and the Brahmaputra of Assam, bursts through the main axis of the range.

At the opposite or northwestern angle, the Indus in like manner pierces the Himalayas, and turns southwards on its course through the Punjab.

The highest point is Dodabetta Peak (8760 ft.), at the upheaved southern angle.

For it is easy to understand by the canons above mentioned that the greatest objects may appear exceedingly small, and the contrary, also that the most remote objects may appear just at hand, and the converse; for we can give such figures to transparent bodies, and dispose them in such order with respect to the eye and the objects, that the rays shall be refracted and bent towards any place we please, so that we shall see the object near at hand or at any distance under any angle we please.

And thus from an incredible distance we may read the smallest letters, and may number the smallest particles of dust and sand, by reason of the greatness of the angle under which we see them....

But, having ascertained by experiment that for all colours of light the angle of incidence is equal to the angle of reflexion, he turned his attention to the construction of reflecting telescopes.

The mere size of the apparent field of view depends upon obtaining the oblique pencils of light emerging from it to cross the axis at the great possible angle, and to this end the presence of a field-lens is indispensable, which is separated from the eye-lens by a considerable interval.

A vertical plane passing through A A is therefore in the meridian, and the polar axis is inclined to the horizon at an angle equal to that of the latitude of the place of observation.

If we now attach to the polar axis a graduated circle D D, called the" hour circle,"of which the microscope or vernier R reads o h when the declination axis is horizontal, we can obviously read off the hour angle from the meridian of any star to which the telescope may be directed at the instant of observation.

If the local sidereal time of the observation is known, the right ascension of the star becomes known by adding the observed hour angle to the sidereal time if the star is west of the meridian, or subtracting it if east of the meridian.

Through the eyepiece of the bent 1 telescope E' another hour circle attached to the lower end of the polar axis can be seen; thus an assistant is able to direct the telescope by a handle at H to any desired hour angle.

The difficulty is that the automatic motion of a single mirror capable of reflecting the rays of any star continuously along the axis of a fixed horizontal telescope, requires a rather complex mechanism owing to the variation of the angle of reflexion with the diurnal motion.

If a mirror is mounted on a truly adjusted polar axis, the plane of the mirror being parallel to that axis, the normal to that mirror will always be directed to some point on the celestial equator through whatever angle the axis is turned.

Besides these complications there is another drawback to the use of the coelostat for general astronomical work, viz., the obliquity of the angle of reflection, which can never be less than that of the declination of the star, and may be greater to any extent.

The sable and roan antelopes are distinguished from Oryx by the stout and thickly ringed horns rising vertically from a ridge over the eyes at an obtuse angle to the plane of the lower part of the face, and then sweeping backwards in a bold curve.

The determination of stellar parallaxes is a matter of great difficulty on account of the minuteness of the angle to be measured, for in no case does the parallax amount to I"; moreover, there is always an added difficulty in determining an annual change of position, for seasonal instrumental changes are liable to give rise to a spurious effect which will also have an annual period.

Now the position of this line, as found by Kobold, actually is a (properly weighted) mean between the corresponding lines of symmetry of the two drifts, but naturally it lies in the acute angle between them, whereas the line of the solar motion is also a weighted mean between the two lines of drift, but lies in the obtuse angle between them.

To turn a line through a certain angle in a given plane, a certain operator is required; but when we wish to turn it through an equal negative angle we must not, in general, employ the negative of the former operator.

For the negative of the operator which turns a line through a given angle in a given plane will in all cases produce the negative of the original result, which is not the result of the reverse operator, unless the angle involved be an odd multiple of a right angle.

The directed line whose length is a, and which makes an angle 0 with the real (positive) unit line, is expressed by a (cos 0+i sin e), where i is regarded as +1,/ - 1.

For this equation merely states that m turnings of a line through successive equal angles, in one plane, give the same result as a single turning through m times the common angle.

We may state, in passing, that every quaternion can be represented as a (cos 0+ 7 sin 9), - where a is a real number, 6 a real angle, and it a directed unit line whose square is - 1.

Also, if 0 be the angle between them, and x", y", z" the direction-cosines of a line perpendicular to each of them, we have xx' +yy'+zz' =cos 0, yz' - zy" = x" sin 0, &c., so that the product of two unit lines is now expressed as - cos0+ (ix" +jy" +kz") sin 0.

Applied to any directed line, this operator at once turns it, conically, through a definite angle, about a definite axis.

The head is covered with a turban, or a cap of a fashion peculiar to the Parsees; it is made of stiff material, something like the European hat, without any rim, and has an angle from the top of the forehead backwards.

As the declination changes the spot of light reflected from the magnet mirror moves parallel to the axis of the recording drum, and hence the distance between the line traced by this spot and the base line gives, for any instant, on an arbitrary scale the difference between the declination and a constant angle, namely, the declination corresponding to the base line.

The value of this constant angle is obtained by comparing the record with the value for the declination as measured with a magnetometer.

To overcome this difficulty Eschenhagen in his earlier type of instruments attached to each magnet two mirrors, their planes being inclined at a small angle so that when the spot reflected from one mirror goes off the paper, that corresponding to the other comes on.

This would bring it into connexion with the Phoenician W (Shin), which, turned through a right angle, is possibly the Greek, though some forms of Zade on old Hebrew coins and gems equally resemble the Greek letter.

In the winter1876-1877a disease which appears to have been plague appeared in two villages in the extreme north of the province of Khorasan, about 25 leagues from the south-east angle of the Caspian Sea.

In March 1877 plague broke out in Resht, a town of 20,000 inhabitants, in the province of Ghilan, near the Caspian Sea at its south-west angle, from which there is a certain amount of trade with Astrakhan.

This makes the Euphrates the main eastern limit, with radii to the north-east angle of the Levant and the south-east angle of the Black Sea, and roughly agrees with the popular conception of Asia Minor as a geographical region.

Before use the strip is for one moment connected to the case, and the arrangement is then tilted until the strip extends at a certain angle.

There is a particular angle of tilt of the case which gives a maximum sensitiveness.

While, however, it thus appears that the mythic part of the Beowulf story is a portion of primeval Angle tradition, there is no proof that it was originally peculiar to the Angles; and even if it was so, it may easily have passed from them into the poetic cycles of the related peoples.

For this reason X is also known as the angle of repose.

If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are X, u, and if the axes are simultaneously turned through an angle e, the coordinates of a point of the lamina, relative to the original axes, are changed from x, y to X+x cos ey sin e, u+x sin e+y cos e, or X + x ye, u + Xe + y, ultimately.

Let a, $ be the inclinations of the planes, and 0 the angle which the bar makes with thE vertical.

If the forces are all parallel, making say an angle 0 with Ox, we may write Xi = Pi cos 0, Vi = P1 sin 0, Xi = P2 cos 0, VI = Pi sin 0,.

This is the axis of the required screw; the amount of the translation is measured by the projection of AB or BC or CD on the axis; and the angle of rotation is given, by the inclination of the aforesaid bisectors.

Successive halfturns about intersecting axes a, b are equivalent to a rotation about the common perpendicular to a, b at their intersection, Of amount equal to twice the acute angle between them, in the direction from a to b.

Successive half-turns about two skew axes a, b are equivalent to a twist about a screw whose axis is the common perpendicular to a, b, the translation being double the shortest distance, and the angle of rotation being twice the acute angle between a, b, in the direction from a to b.

An infinitely small rotation about any axis is conveniently represented geometrically by a length AB measures along the axis and proportional to the angle of rotation, with the convention that the direction from A to B shall be related to the rotation as is the direction of translation to that of rotation in a righthanded screw.

The work of a couple in any infinitely small rotation of a rigid body about an axis perpendicular to the plane of the couple is equal to the product of the moment of the couple into the angle of rotation, proper conventions as to sign being observed.

Draw JBA perpendicular to the lines of action, and let e be the angle of rotation.

If the body be supposed to roll - (say to the right) until the curves touch at J, and if JJ=bs, the angle through which the u,pper figure rotates is Is/p +Is/p, and the horizontal displace- V ment, of G is equal to the product of this expression into h.

If these lines be turned through a right angle in the same sense, they take up positions such as CC, DID, where C, D are on JC, JD, respectively, and CD is parallel to CD.

The instantaneous centre of CD will be at the intersection of AD, BC, and if CD be drawn parallel to CD, the lines CC, DD may be taken to represent the virtual velocities of C, D turned each through B a right angle.

The angle ot+e (or AOQ) is called the phase; the arbitrary FIG.

The orbit has therefore two asymptotes, inclined at an angle lr/m.

It follows that the angle between successive apse-lines is constant; it is called the apsidal angle of the orbit.

If this is the case, the apsidal angle must evidently be commensurable with -ir, and since it cannot vary discontinuously the apsidal angle in a nearly circular orbit must be constant.

In any case of a central orbit the hodograph (when turned through a right angle) is similar and similarly situated to the reciprocal polar of the orbit with respect to the centre of force.

As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position.

When the body is twisted through an angle 0 the threads make angles aC/i, be/I with the vertical, and the moment of the tensions about the vertical through G is accordingly Kg, where K = M gab/i.

Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular Cartesian co-ordinates x, y of the mass-centre G and the angle C through which the body has turned from some standard position.

To find the small oscillation about a state of steady precession in which the axis makes a constant angle a with the vertical, we write O=a+X, and neglect terms of the second order in x.

The result is that the axis of the top describes a circular cone about a fixed line making a small angle with the vertical.

Let r be the distance of a point P from a fixed origin 0, 0 the angle which OP makes with a fixed direction OZ, il the azimuth of the plane ZOP relative to some fixed plane through OZ.