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tangent

The distance of the lucid points was the tangent of the magnified angles subtended by the stars to a radius of io ft.

Fermat and Descartes agreed in regarding the tangent to a curve as a secant of that curve with the two points of intersection coinciding, while Roberval regarded it as the direction of the composite movement by which the curve can be described.

The tangent at any point bisects the angle between the focal distances of the point, and the normal is equally inclined to the focal distances.

Also the auxiliarly circle is the locus of the feet of the perpendiculars from the foci on any tangent.

If the tangent at P meets the asymptotes in R, R', then CR.CR' = CS 2.

Between the Andamans and Cape Negrais intervene two small groups, Preparis and Cocos; between the Andamans and Sumatra lie the Nicobar Islands, the whole group stretching in a curve, to which the meridian forms a tangent between Cape Negrais and Sumatra; and though this curved line measures 700 m., the widest sea space is about 91 m.

The smoothest and safest running is, in fact, attained when a " transition," " easement " or " adjustment " curve is inserted between the tangent and the point of circular curvature.

They consist of a long rectangular building, with a proscenium or column front which almost forms a tangent to the circle of the orchestra; at the middle and at either end of this proscenium are doors leading into the orchestra, those at the end set in projecting wings; the top of the proscenium is approached by a ramp, of which the lower part is still preserved, running parallel to the parodi, but sloping up as they slope down.

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.

The intrinsic equation, expressing the relation between the arc 0- (measured from 0) and the inclination 4) of the tangent at any points to the axis of x, assumes a very simple form.

The cartesian equation is x = ti' (c2-y'")+ 2c log [{c-?/ (c.2- y2)}/{c+?i (c2+y2)il, and the curve has the geometrical property that the length of its tangent is constant.

Putting (12) a vortex line is defined to be such that the tangent is in the direction of w, the resultant of, n, called the components of molecular rotation.

(io) The velocity q is zero in a corner where the hyperbola a cuts the ellipse a; and round the ellipse a the velocity q reaches a maximum when the tangent has turned through a right angle, and then q _ (Ch 2a-C0s 2(3).

Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, px _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p d = ds; (8) Thence d?

This mass is equal to 47rabcp,u; therefore Q = A47rabcp s and b =pp, where p is the length of the perpendicular let fall from the centre of the ellipsoid on the tangent plane.

Accordingly for a given ellipsoid the surface density of free distribution of electricity on it is everywhere proportional to the the tangent e plane e att that point.

all from thhiseweecan of determine the capacity of the ellipsoid as follows: Let p be the length of the perpendicular from the centre of the ellipsoid, whose equation is x 2 /a 2 -1-y2/b2 -1-,2c2 = i to the tangent plane at x, y, z.

The diameter of a quadric surface is a line at the extremities of which the tangent planes are parallel.

The width of each of the portions aghc and acfe cut away from the lens was made slightly greater than the focal length of lens X tangent of sun's greatest diameter.

Here, in order to fulfil the purposes of the previous models, the distance of the centres of the lenses from each other should only slightly exceed the tangent of sun's diameter X focal length of lenses.

9) by the tangent screw f, acting on a small toothed wheel clamped to the rod connected with the driving pinion, there was apt to be a torsion of the rod rather than an immediate action.

This wheel is acted on by a tangent screw whose bearings are attached to the cradle; the screw is turned by means of a handle supported by bearings attached to the cradle, and coming within convenient reach of the observer's hand.

The slowest speed is given by means of a tangent screw which is carried by a ball-bearing on the flange of the telescope sleeve, whilst its nut is double-jointed to a ring that encircles the flange of the heliometer-tube.

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.

- Dispart and Tangent Sights.

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

The earliest form of a hind or breech sight was fixed, but in the early part of the 19th century Colonel Thomas Blomefield proposed a movable or tangent sight.

It was not, however, till 1829 that a tangent sight (designed by Major-General William Millar) was introduced into the navy; this was adopted by the army in 1846.

As the tangent sight was placed in the line of metal, hence directly over the cascable, very little movement could be given to it, so that a second sight was required for long ranges.

i it will be seen that in order to strike T the axis must be directed to G' at a height above T equal to TG, while the line of sight or line joining the notch of the tangent sight and apex of the dispart or foresight must be (/ ?` directed on T.

4 the tangent sight has been raised from 0 to S, the line of.

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.

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

In practice, tangent sights were graduated graphically from large scale drawings.

Tangent sights were not much trusted at first.

could be done by noting the amount of deflection for each range and applying it by means of a sliding leaf carrying the notch, and it is so done in howitzers; in most guns, however, it is found more convenient and sufficiently accurate to apply it automatically by inclining the socket through which the tangent scale rises.

- Theory of Tangent Sight.

Other improvements were: the gun was sighted on each side, tangent scales dropping into sockets in a sighting ring on the breech, thus enabling a long scale for all ranges to be used, and the foresights screwing into holes or dropping into sockets in the trunnions, thus obviating the fouling of the line of sight, and the damage to FIG.

The tangent sight was graduated in yards as well as degrees and had also a fuze scale.

- Laying by Full tangent sight, the point of the fore g y sight and the target must be in line " Sight.

Since the early days of rifled guns tangent sights have been improved in details, but the principles remain the same.

Except for some minor differences the tangent sights were the same for all natures of guns, and for all services, but the development of the modern sight has followed different lines according to the nature and use of the gun, and must be treated under separate heads.

With the exception of the addition of a pin-hole to the tangent sight and cross wires to the fore-sight, and of minor improvements, and Field of the introduction of French's crossbar sight and the artillery reciprocating sight, of which later, no great advance was sights.

The disadvantages that still remain are that the sight has to be removed every time the gun is fired, and the amount of deflection is limited and has to be put on the reverse way to that on a tangent scale.

The normal method of laying these is from the fore-sight over the tangent sight to a point in rear.

16) is that the tangent sight has a steel horizontal bar which can slide through the head of the tangent scale for deflection, and is graduated for 3° left and 1 ° right deflection.

At night this mark is replaced by a lamp installed in rear sight has a fixed horizontal bar slotted and graduated similarly to the slotted portion of the tangent sight.

The leaf of the fore-sight has a pinhole, and that of the tangent sight cross-wires for fine reverse laying.

Fore-sights are made right and left; tangent sights are interchangeable, the graduations are cut on the horizontal edges above and below, so that the sight can be changed from right to left or vice versa by removing and reversing the bar.

The advantages compared with a tangent sight are that only half the movement is required to raise the sight for any particular range; the ranges on the drum are easier to read, and if necessary can be set by another man, so that the layer need not take his eye from the telescope.

19) represent a gun at height BD above water-level DC, elevated to such an angle that a shot would strike the water at C. Draw EB parallel to DC. It is clear that under these conditions, if a tangent sight AF be raised to a height F representing the elevation due to the range BC, the object C will be on the line of sight.

The tangent scale moved freely in a socket fixed to the gun; its lower end rested on one of the cams, cut to a correct curve.

In the navy the conditions of an unstable platform rendered quadrant elevation of little use, and necessitated a special pattern of tangent sight to facilitate firing the moment the roll of the ship brought the sights on the target.

The fore-sight was a small globe, and in the original patterns this was placed on a movable leaf on which deflection for speed of one's own ship was given, while deflection for speed of enemy's ship and wind were given on the tangent sight.

In subsequent patterns all the deflection was given on the tangent sight, which was provided with two scales, the upper one graduated in knots for speed of ship, and the lower one in degrees.

Draw the tangents at A and B, meeting at T; draw TV parallel to the axis of the parabola, meeting the arc in C and the chord in V; and M draw the tangent at C, meeting AT and BT in a and b.

The fact that C 1 does not give the true area is due to the fact that in passing from one extremity of the top of any strip to the other extremity the tangent to the trapezette E, _- changes its direction.

Hence, if the angle which the tangent at the extremity of the ordinate u 0 makes with the axis of x is denoted by fie, we have area from uo to u1= 2h(uo + ui) - -- i i h 2 (tan y l - tan t u 2 = Wu ' + u2) - 1 Tih 2 (tan 4,2 - tan um-1 t0 26 m, - 2 h(um-1 + um) i h (tan 4, m - and thence, by summation, A =C I - i i h 2 (tan - tan 1,1/o).

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.

12 represent a horizontal section of the dome through the source P. Let OPA be the radius through P. Let PQ represent a ray of sound making the angle B with the tangent at A.

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

The tangent to the displacement curve is always parallel to the axis, that is, for a small distance the successive particles are always equally displaced, and therefore always occupy the same volume.

So also there is on the whole none in that direction leaving at P. Let the tangent at P make angle 4) with AB.

But the tension at P is T, parallel to the tangent, and T sin 4 parallel to PM, and through this - T sin is the momentum passing out at P per second.

that there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts, the four spheres at right angles; this "orthotomic" sphere corresponds to the orthogonal circle of a system of circles.

70, at a distance x from the vertex, the horizontal component of the resultant (tangent to the curve) will be unaltered; the vertical component V will be simply the sum of the loads between 0 and F, or wx.

In the triangle FDC, let FD be tangent to the curvelFC vertical, and Dqhorizontal; these three sides will necessarily be proportional respectively.to_the FIG.

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..

72 with arcs of the length 1,, l2, l3, &c., and with the radii r1, r 2, &c. (note, for a length 2l 1 at each end the radius will be infinite, and the curve must end with a straight line tangent to the last arc), then let v be the measured deflection of this curve from the straight line, and V the actual deflection of the bridge; we have V = av/b, approximately.

This value of is the tangent elevation (T.E); the quadrant elevation (Q.E.) is -S, where S is the angular depression of the line of sight and if 0 is h ft.

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.

Now taking equation (72), and replacing tan B, as a variable final tangent of an angle, by tan i or dyldx, (75) tan 4) - dam= C sec n [I(U) - I(u)], and integrating with respect to x over the arc considered, (76) x tan 4, - y = C sec n (U) - f :I(u)dx] 0 But f (u)dx= f 1(u) du = C cos n f x I (u) u du g f() =C cos n [A(U) - A(u)] in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference AA, where (78) AA = I (u) 9 = I (u) or else by an integration when it is legitimate to assume that f(v) =v m lk in an interval of velocity in which m may be supposed constant.

DG, it is a " tangent."

The equations to the chord, tangent and normal are readily derived by the ordinary methods.

5) be the points of contact of a common tangent; drop perpen FIG.

To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle.

The radical axis is x = o, and it may be shown that the length of the tangent from a point (o, h) is h 2 k 2, i.e.

and produced it to meet the tangent at A in E; and then his assertion (not established by him) was that AE was nearly equal to the arc AC, the error being in defect.

II) whose centre is 0, AC its chord, and HK the tangent drawn at the middle point of the arc and bounded by OA, OC produced, then, according to Archimedes, AMC AC. In modern trigonometrical notation the propositions to be compared stand as follows: 2 tan 20 >2 sin 28 (Archimedes); tan 10+2 sin 3B>0> 3 sin B (Snell).

As far as the circlesquaring functions are concerned, it would seem that Gregory was the first (in 1670) to make known the series for the arc in terms of the tangent, the series for the tangent in terms of the arc, and the secant in terms of the arc; and in 1669 Newton showed to Isaac Barrow a little treatise in manuscript containing the series for the arc in terms of the sine, for the sine in terms of the arc, and for the cosine in terms of the arc. These discoveries 1 See Euler, ” Annotationes in locum quendam Cartesii," in Nov.

If this be applied to the right-hand side of the identity m m m 2 m2 tan-=- - n n -3n-5n" it follows that the tangent of every arc commensurable with the radius is irrational, so that, as a particular case, an arc of 45 having its tangent rational, must be incommensurable with the radius; that is to say, 3r/4 is an incommensurable number."

The generality of treatment is indeed remarkable; he gives as the fundamental property of all the conics the equivalent of the Cartesian equation referred to oblique axes (consisting of a diameter and the tangent at its extremity) obtained by cutting an oblique circular cone in any manner, and the axes appear only as a particular case after he has shown that the property of the conic can be expressed in the same form with reference to any new diameter and the tangent at its extremity.

Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic.

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.

It may be defined as a section of a right circular cone by a plane parallel to a tangent plane to the cone, or as the locus of a point which moves .so that its distances from a fixed point and a fixed line are equal.

parallel to the tangent at the vertex of the diameter and is equal P A B to four times the focal distance of the vertex.

property that the line at infinity is a tangent.

The simplest equation to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of co-ordinates, when it assumes the form y 2 = 4ax where as = semilatus rectum; this may be deduced directly from the definition.

An equation of similar form is obtained when the axes of co-ordinates are any diameter and the tangent at the vertex.

The tangent then becomes my=x+amt and the normal y +aam - am 3 .

9, the axis of x being a double tangent.

should be converted economical method, but, as will be seen in the diagram, the quality of the boards will vary very much, some consisting almost entirely of sap-wood cut at a tangent to the annular rings such as a, b, c, whilst the centre boards contain the heartwood cut in the best way at right angles across the annual rings as d, e, f.

the line parallel to q' q-- 1 which intersects the axes of Q and Q'; the plane of the member contains a fixed line; the centre is on a fixed ellipse which intersects the transversal; the axis is on a fixed ruled surface to which the plane of the ellipse is a tangent plane, the ellipse being the section of the ruled surface by the plane; the ruled surface is a cylindroid deformed by a simple shear parallel to the transversal.

This was a digression of a new kind, if anything can be called a digression in a work the plan of which is to fly off at a tangent whenever and wherever the writer's whim tempts him.

It appears also from (II) that the null-lines whose distance from the central axis is r are tangent lines to a system of helices of slope tan 1(r/k); and it is to be noticed that these helices are left-handed if the given wrench is righthanded, and vice versa.

where p, p are the radii of curvature of the two curves at J, 4~ is the inclination of the common tangent at J to the horizontal, and h is the height of G above J.

It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element s must satisfy the conditions of equilibrium laid down in I.

~/, (= s) in the directions of the tangent and normal respectively.

-- T.r-OT If T, T + aT be the tensions at P, Q, and 4 be the angle between the directions of the curve at these points, the components Q of the tensions along the tangent at P give (T + T) cos T,

and therefore varies as the square of the perpendicular drawn from 0 to a tangent plane of a certain quadric surface, the tangent plane in question being parallel to (22).

If the co-ordinate axes coincide with the principal axes of this quadric, we shall have ~(myz) =0, ~(mzx) =0, Z(mxy) = 0~ (24) and if we write ~(mx) = Ma, ~(my1) = Mb, ~(mz) =Mc2, (25) where M=~(m), the quadratic moment becomes M(aiX2+bI,s2+ cv), or Mp, where p is the distance of the origin from that tangent plane of the ellipsoid ~-,+~1+~,=I, (26)

Now consider the tangent plane w at any point P of a confocal, the tangent plane fii at an adjacent point N, and a plane of through P parallel to of.

If we write A=Ma, B=M/32, C=M~y, the formula (37), when referred to the principal axes at 0, becomes if p denotes the perpendicular drawn from 0 in the direction (X, u, e) to a tangent plane of the ellipsoid ~+~+~=I (43)

Obviously OV is parallel to the tangent to the path atP, and its magnitude is ds/dt, where s is the arc. If we project OV on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections u, v, w are called the component velocities parallel to the axes.

In the time iSt the velocity parallel to the tangent at p P changes from v to v+v, ulti- ~FIG.

If the axes of x and y be drawn horizontal and vertical (upwards), and if ~ be the inclination of the tangent to the horizontal, we have dv.

In symbols, if v be the velocity and p the perpendicular from 0 to the tangent to the path, pv=h, (1)

But since an equiangular spiral having a given pole is completely determined by a given point and a given tangent, this type of orbit is not a general one for the law of the inverse cube.

S is the centre of force, SY is the per- z pendicular to the tangent at P, and Z is the point where VS meets the auxiliary circle again.

Now the moment of this localized vector with respect to any axis through G is zero, to the first order of &, since the perpendicular distance of G from the tangent line at G is of the order (ot)2.

The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis.

the length of the perpendicular OH on the tangent plane at J

We have seen (~ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that ~ (2)

Since w varies as p, it follows that OH is constant, and the tangent plane at J is therefore fixed in space.

The components -.; of the reaction of the horizontal, lane will be Mc~f at right angles LI ~-~: to the tangent line at the point Mc~2 - of contact and Mg vertically up wards, and the moment of these FIG.

The angle whose tangent is the coefficient of friction is called thf angle of repose, and is expressed symbolically by 4) tan_if.

a straight tangent to the pitch-circle at that point; R the internal and R the equal external describing circles, so placed as to touch the pitch-circle and each other at I.

The total pressure exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity, or inclination to the common perpendicular of the surfaces, is the angle of repose formerly mentioned in 14, whose tangent is the coefficient of friction.

Now let PT be a tangent to the curve at P, cutting OX in T; PT=PYXsecant obliquity, and this is to be a constant quantity; hence the curve is that known as the tractory of the straight line OX, in which PT = OR = constant, This curve is described by having a fixed straight edge parallel to OX, along which slides a slider carrying a pin whose centre is T.

The moment of friction of Schieles anti-friction pivot, as it is called, is equal to that of a cylindrical journal of the radius OR=PT the constant tangent, under the same pressure.

Pouillet in 1837 contributed the sine and tangent compass, and W.

von Helmholtz devised a tangent galvanometer with two coils.

In the neighbourhood of 550 pu the tangent to the curve is parallel to the axis of wave-lengths; and the focal length varies least over a fairly large range of colour, therefore in this neighbourhood the colour union is at its best.

The three angles between the tangent planes to the three surfaces of separation at the point 0 are completely determined by the tensions of the b o a three surfaces.

If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.

Let the radius of this section PR by y, and let PT, the tangent at P, make an angle a with the axis.

Also, since the axis is a tangent to the rolling curve, the ordinate PR is the perpendicular from the tracing point P on the tangent.

Hence the relation between the radius vector and the perpendicular on the tangent of the rolling curve must be identical with the relation between the normal PN and the ordinate PR of the traced curve.

If we write r for PN, then y= r cos a, and equation 9 becomes 13.7,T - I) This relation between y and r is identical with the relation between the perpendicular from the focus of a conic section on the tangent at a given point and the focal distance of that point, provided the transverse and conjugate axes of the conic are 2a and 2b respectively, where a= p, and b 2 = -.

If K is the height of the flat surface of the drop, and k that of the point where its tangent plane is vertical, then T = 1(K - k) 2gp. Quincke finds that for several series of substances the surfacetension is nearly proportional to the density, so that if we call Surface-Tensions of Liquids at their Point of Solidification.

But his contempt for the annalistic form makes him at times careless in his chronology and arbitrary in his method of arranging his material; he not infrequently flies off at a tangent to relate stories which have little or no connexion with the main narrative; his critical faculty is too often allowed to lie dormant.

At the same time he challenged Roberval and Fermat to construct the tangent; Roberval failed but Fermat succeeded.

The b co-ordinates of any point R on the a ?/' t11®V1 a cycloid are expressible in the form x=a(8-}-sin 0); y=a (I -cos 0), M where the co-ordinate axes are the tangent at the vertex 0 and the axis of the curve, a is the radius of the generating circle, and 0 the angle R'CO, where RR' is parallel to LM and C is the centre of the circle in its symmetric position.

The cartesian equation, referred to the fixed diameter and the tangent at B as axes may be expressed in the forms x= a6, y=a(I -cos 0) and y-a=a sin (x/afir); the latter form shows that the locus is the harmonic curve.

Barrow Tangent Pt.

F B is the evolute of this circle, and for any radius DE at an angle a and corresponding tangent EG terminated by the evolute, the perpendicular distance of G from the line AD is c(cos a+a sin a).

He found that the moon by her motion in her orbit was deflected from the tangent in every minute of time through a space of thirteen feet.

The Greek geometers invented other curves; in particular, the conchoid, which is the locus of a point such that its distance from a given line, measured along the line drawn through it to a fixed point, is constant; and the cissoid, which is the locus of a point such that its distance from a fixed point is always equal to the intercept (on the line through the fixed point) between a circle passing through the fixed point and the tangent to the circle at the point opposite to the fixed point.

Plucker first gave a scientific dual definition of a curve, viz.; " A curve is a locus generated by a point, and enveloped by a line - the point moving continuously along the line, while the line rotates continuously about the point "; the point is a point (ineunt.) of the curve, the line is a tangent of the curve.

A line has only a point-equation, and a point has only a lineequation; but any other curve has a point-equation and also a line-equation; the point-equation (* x, y, z) m = o is the relation which is satisfied by the point-co-ordinates (x, y, z) of each point of the curve; and similarly the line-equation (*, 1 7, 0"= o is the relation which is satisfied by the line-co-ordinates (E, r7,) of each line (tangent) of the curve.

The stationary tangent: the line may in the course of its rotation come to rest, and then reverse the direction of its rotation.

The double tangent: the line may in the course of its motion come to coincide with a former position of the line, the two positions of the point not in general coinciding.

It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or acnode), nor again the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to.

for the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangent and inflections.

But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution and for a cusp a diminution 3, in the number of the intersections; and thus, for a curve with 6 nodes and K cusps, there is a diminution 26+3K, and the value of n is n= m (m - I)-26-3K.

The most simple case is when three double points come into coincidence, thereby giving rise to a triple point; and a somewhat more complicated one is when we have a cusp of the second kind, or node-cusp arising from the coincidence of a node, a cusp, an inflection, and a double tangent, as shown in the annexed figure, which represents the singularities as on the point of coalescing.

Mag., 1858; considering the m - 2 points in which any tangent to the curve again meets the curve, he showed how to form the equation of a curve of the order (m - 2), giving by its intersection with the tangent the points in question; making the tangent touch this curve of the order (m - 2), it will be a double tangent of the original curve.

At any one of the m 2 -26 - 3K points the variable curve and the consecutive curve have tangents distinct from yet infinitesimally near to each other, and each of these two tangents is also infinitesimally near to one of the n 2 -2T-3t common tangents of the two curves; whence, attending only to the variable curve, and considering the consecutive curve as coming into actual coincidence with it, the n 2 -2T-3c common tangents are the tangents to the variable curve at the m 2 -26-3K points respectively, and the envelope is at the same time generated by the m 2 -26-3K points, and enveloped by the n2-2T-3c tangents; we have thus a dual generation of the envelope, which only differs from Pliicker's dual generation, in that in place of a single point and tangent we have the group of m2-26-3K points and n 2 -2T-3c tangents.

Many well-known derivative curves present themselves in this manner; thus the variable curve may be the normal (or line at right angles to the tangent) at any point of the given curve; the intersection of the consecutive normals is the centre of curvature; and we have the evolute as at once the locus of the centre of curvature and the envelope of the normal.

Names may also be used for the different forms of infinite branches, but we have first to consider the distinction of hyperbolic and parabolic. The leg of an infinite branch may have at the extremity a tangent; this is an asymptote of the curve, and the leg is then hyperbolic; or the leg may tend to a fixed direction, but so that the tangent goes further and further off to infinity, and the leg is then parabolic; a branch may thus be hyperbolic or parabolic as to its two legs; or it may be hyperbolic as to one leg and parabolic as to the other.

If a line S2 cut an arc aa at b, so that the two segments ab, ba lie on opposite sides of the line, then projecting the figure so that the line Sl goes off to infinity, the tangent at b is projected into the asymptote, and the arc ab is projected into a hyperbolic leg touching the asymptote at one extremity; the arc ba will at the same time be projected into a hyperbolic leg touching the same asymptote at the other extremity (and on the opposite side), but so that the two hyperbolic legs may or may not belong to one and the same branch.

Next, if the line S2 touch at b the arc aa so that the two portions ab, ba lie on the same side of the line Sl, then projecting the figure as before, the tangent at b, that is, the line S2 itself, is projected to infinity; the arc ab is projected into a parabolic leg, and at the same time the arc ba is projected into a parabolic leg, having at infinity the same direction as the other leg, but so that the two legs may or may not belong to the same branch.

It will readily be understood how the like considerations apply to other cases, - for instance, if the line is a tangent at an inflection, passes through a crunode, or touches one of the branches of a crunode, &c.; thus, if the line S2 passes through a crunode we have pairs of hyperbolic legs belonging to two parallel asymptotes.

Thirdly, the three intersections by the line infinity may be coincident and real; or say we have a threefold point: this may be an inflection, a crunode or a cusp, that is, the line infinity may be a tangent at an inflection, and we have the divergent parabolas; a tangent at a crunode to one branch, and we have the trident curve; or lastly, a tangent at a cusp, and we have the cubical parabola.

It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent; consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none, - f our if the point is on the odd circuit, none if it is on the even circuit.

A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is =4; it follows that the quartic has at most 8 real inflections.

Again, the normal, qua line at right angles to the tangent, is connected with the circular points, and these accordingly present themselves in the before-mentioned theories of evolutes and parallel curves.

Such a curve may be considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the same time rotates about the line; the point is a point, the line a tangent, and the plane an osculating plane, of the curve; moreover the line is a generating line, and the plane a tangent plane, of a developable surface or torse, having the curve for its edge of regression.

A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the intersections of the tangents from the circular points at infinity; the centre of a conic the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear co-ordinates.

Considered as a curved surface, concentric with the earth, a tangent plane to such a surface is the plane of the horizon.

Let APB be a semicircle, BT the tangent at B, and APT a line cutting the circle in and BT at T; take a point Q on AT so that AQ always equals PT; then the locus of Q is the cissoid.

Of the properties of a tangent it may be noticed that the tangent at any point is equally inclined to the focal distances of that point; that the feet of the perpendiculars from the foci on any tangent always lie on the auxiliary circle, and the product of these perpendiculars is constant, and equal to the product of the distances of a focus from the two vertices.

The middle points of a system of parallel chords is a straight line, and the tangent at the point where this line meets the curve is parallel to the chords.

The equation to the tangent at 0 is x cos 0/a+y sin 0/b = 1, and to the normal ax/cos 0 - by/sin 0=a2 - b'.

The principle of involution solves such constructions as: given four tangents and one point, three tangents and two points, &c. If a tangent and its point of contact be given, it is only necessary to remember that a double point on the curve is given.

A focus or directrix is equal to two conditions; hence such problems as: given a focus and three points; a focus, two points and one tangent; and a focus, one point and two tangents are soluble (very conveniently by employing the principle of reciprocation).

And so by coming into connexion with different reals the "self-preservations" of A will vary accordingly, A remaining the same through all; just as, by way of illustration, hydrogen remains the same in water and in ammonia, or as the same line may be now a normal and now a tangent.

The polarizing angle varies from one transparent substance to another, and Sir David Brewster in 1815 enunciated the law that the tangent of the polarizing angle is equal to the refractive index of the substance.

and the distance 1, which is equal to the tangent of the visual angle w, is termed the " apparent size " of the object.

The ratio of half the length of the visible piece of the scale to its distance from the diaphragm on the stage gives the tangent of half the angular aperture.

approximate the tangent function by its small argument expansion.

Upper and lower tangent arcs and a faint 22º halo.

As the sun climbs the upper tangent arc opens and then droops like a gull's wings.

But then he'll veer off at a tangent, or throw in a fart gag and the feeling is lost.

This is because the gradient of a curve at a point is equal to the gradient of the tangent at that point.

Jacobian spectra for both and find tangent altitude range where these are distinguishable.

In addition to covering the cases above it extends the method to bodies with paraboloidal, or blunted tangent ogive or blunted conical forebodies.

They use sine, cosine and tangent in right-angled triangles when solving problems in two dimensions.

Using trigonometry: angles and the trigonometric functions sine, cosine and tangent; usage in the Earth and the science of surveying.

tangent of the angle of the sun from the wall.

He also shows how to draw a tangent to three given lines.

The leftmost column is column 0. TAN A function returning the tangent of its radian argument.

The external circle of the tower has a perfect tangent in the nave west wall.

The following diagram illustrates the calculation using the tangent of the angle.

So, all we need to do is construct the tangent and measure its gradient, Δ y / Δ x.

The inverse tangent is the value whose tangent is ` x ' .

Parameters: a - the value whose arc tangent is to be returned.

OK now let us go off at a slight tangent for a few moments.

You can use these pages to follow the day from your machine and go off on your own tangent should you wish.

tangent ogive or blunted conical forebodies.

tangent modulus is given.

tangent arc was the bright area on the horizon directly below the sun.

tangent screw.

tangent altitudes unchanged this may be as a result of change in ILS?

tangent planes make an angle of.

Measurements of permitivity and loss tangent as a function of temperature and bias are obtained in this way.

Don't go off on a tangent or get too verbose.

The edges are then separated till they are tangent to the opposite limbs of the disk of the planet to be measured, or till they respectively bisect two stars, the angle between which is to be determined.

The distance of the lucid points was the tangent of the magnified angles subtended by the stars to a radius of io ft.

Here and there particular curves, for example, had been obliged to yield the secret of their tangent; but the ancient geometers apparently had no consciousness of the general bearings of the methods which they so successfully applied.

Fermat and Descartes agreed in regarding the tangent to a curve as a secant of that curve with the two points of intersection coinciding, while Roberval regarded it as the direction of the composite movement by which the curve can be described.

The circle on AA' as diameter is called the auxiliarly circle; obviously AN.NA' equals the square of the tangent to this circle from N, and hence the ratio of PN to the tangent to the auxiliarly circle from N equals the ratio of the conjugate axis to the transverse.

The tangent at any point bisects the angle between the focal distances of the point, and the normal is equally inclined to the focal distances.

Also the auxiliarly circle is the locus of the feet of the perpendiculars from the foci on any tangent.

If the tangent at P meet the conjugate axis in t, and the transverse in N, then Ct.

If the tangent at P meets the asymptotes in R, R', then CR.CR' = CS 2.

Between the Andamans and Cape Negrais intervene two small groups, Preparis and Cocos; between the Andamans and Sumatra lie the Nicobar Islands, the whole group stretching in a curve, to which the meridian forms a tangent between Cape Negrais and Sumatra; and though this curved line measures 700 m., the widest sea space is about 91 m.

Reverse curves are compound curves in which the components are of contrary flexure, like the letter S; strictly the term is only applicable when the two portions follow directly one on the other, but it is sometimes used of cases in which they are separated by a " tangent " or portion of straight line.

The smoothest and safest running is, in fact, attained when a " transition," " easement " or " adjustment " curve is inserted between the tangent and the point of circular curvature.

They consist of a long rectangular building, with a proscenium or column front which almost forms a tangent to the circle of the orchestra; at the middle and at either end of this proscenium are doors leading into the orchestra, those at the end set in projecting wings; the top of the proscenium is approached by a ramp, of which the lower part is still preserved, running parallel to the parodi, but sloping up as they slope down.

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.

The intrinsic equation, expressing the relation between the arc 0- (measured from 0) and the inclination 4) of the tangent at any points to the axis of x, assumes a very simple form.

that at any point the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph equal to the magnitude of the resultant acceleration at the corresponding point of the orbit.

Phil.): - Let x, y, z be the coordinates of P in the orbit,, r t, those of the corresponding point T in the hodograph, then dx dy _ dz c= ' 71 - a' - at therefore Also, if s be the arc of the hodograph, ds = v = V V1 1) j dt + (dt2) dt Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration.

The cartesian equation is x = ti' (c2-y'")+ 2c log [{c-?/ (c.2- y2)}/{c+?i (c2+y2)il, and the curve has the geometrical property that the length of its tangent is constant.

If G was above M, the tangent drawn from G to the evolute of B, and normal to the curve of buoyancy, would give the vertical in a new position of equilibrium.

Putting (12) a vortex line is defined to be such that the tangent is in the direction of w, the resultant of, n, called the components of molecular rotation.

(io) The velocity q is zero in a corner where the hyperbola a cuts the ellipse a; and round the ellipse a the velocity q reaches a maximum when the tangent has turned through a right angle, and then q _ (Ch 2a-C0s 2(3).

Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, px _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p d = ds; (8) Thence d?

This mass is equal to 47rabcp,u; therefore Q = A47rabcp s and b =pp, where p is the length of the perpendicular let fall from the centre of the ellipsoid on the tangent plane.

Accordingly for a given ellipsoid the surface density of free distribution of electricity on it is everywhere proportional to the the tangent e plane e att that point.

all from thhiseweecan of determine the capacity of the ellipsoid as follows: Let p be the length of the perpendicular from the centre of the ellipsoid, whose equation is x 2 /a 2 -1-y2/b2 -1-,2c2 = i to the tangent plane at x, y, z.

The diameter of a quadric surface is a line at the extremities of which the tangent planes are parallel.

The width of each of the portions aghc and acfe cut away from the lens was made slightly greater than the focal length of lens X tangent of sun's greatest diameter.

Here, in order to fulfil the purposes of the previous models, the distance of the centres of the lenses from each other should only slightly exceed the tangent of sun's diameter X focal length of lenses.

9) by the tangent screw f, acting on a small toothed wheel clamped to the rod connected with the driving pinion, there was apt to be a torsion of the rod rather than an immediate action.

This wheel is acted on by a tangent screw whose bearings are attached to the cradle; the screw is turned by means of a handle supported by bearings attached to the cradle, and coming within convenient reach of the observer's hand.

The slowest speed is given by means of a tangent screw which is carried by a ball-bearing on the flange of the telescope sleeve, whilst its nut is double-jointed to a ring that encircles the flange of the heliometer-tube.

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.

- Dispart and Tangent Sights.

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

The earliest form of a hind or breech sight was fixed, but in the early part of the 19th century Colonel Thomas Blomefield proposed a movable or tangent sight.

It was not, however, till 1829 that a tangent sight (designed by Major-General William Millar) was introduced into the navy; this was adopted by the army in 1846.

The tangent sight (see fig.

As the tangent sight was placed in the line of metal, hence directly over the cascable, very little movement could be given to it, so that a second sight was required for long ranges.

i it will be seen that in order to strike T the axis must be directed to G' at a height above T equal to TG, while the line of sight or line joining the notch of the tangent sight and apex of the dispart or foresight must be (/ ?` directed on T.

4 the tangent sight has been raised from 0 to S, the line of.

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.

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

In practice, tangent sights were graduated graphically from large scale drawings.

Tangent sights were not much trusted at first.

could be done by noting the amount of deflection for each range and applying it by means of a sliding leaf carrying the notch, and it is so done in howitzers; in most guns, however, it is found more convenient and sufficiently accurate to apply it automatically by inclining the socket through which the tangent scale rises.

- Theory of Tangent Sight.

amount of left deflection given - the amount can easily be determined thus: The height of tangent scale for any degree of elevation is given with sufficient accuracy by the rough rule for circular measure It= X 1200 where a is the angle of elevation in minutes, h the height 3 of the tangent scale, and R the sighting radius; thus for 10 h _ 60 X R _ 2.0 Now supposing the sight is inclined IÃ‚° to the left, 3600 - which will move the notch from H to H' (see fig.

Other improvements were: the gun was sighted on each side, tangent scales dropping into sockets in a sighting ring on the breech, thus enabling a long scale for all ranges to be used, and the foresights screwing into holes or dropping into sockets in the trunnions, thus obviating the fouling of the line of sight, and the damage to FIG.

The tangent sight was graduated in yards as well as degrees and had also a fuze scale.

- Laying by Full tangent sight, the point of the fore g y sight and the target must be in line " Sight.

Since the early days of rifled guns tangent sights have been improved in details, but the principles remain the same.

Except for some minor differences the tangent sights were the same for all natures of guns, and for all services, but the development of the modern sight has followed different lines according to the nature and use of the gun, and must be treated under separate heads.

The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.