Propositions I-II are preliminary, 13-20 contain tangential properties of the curve now known as the spiral of Archimedes, and 21-28 show how to express the area included between any portion of the curve and the radii vectores to its extremities.
Sometimes the epidermis is considerably more developed by tangential division of its cells, forming a many-layered water-tissue.
2o.Laticiferous vessels from the cortex of the root Scoyzonera hispanica, tangential secf ion.
Sometimes development stops altogether, and a layer of undifferentiated parenchyma (the mesodesm) is left between them; or it may continue indefinitely, the central cells keeping pace by their tangential division with the differentiation of tissue on each side.
New tangential walls arise in the cells which are the seat of cambial activity, and an initial layer of cells is established which cuts off tissue mother-cells on the inside and outside, alternately contributing to the xylem and to the phloem.
The parenchyma is often arranged in tangential bands between the layers of sievetubes and tracheal elements.
The fibres are frequently found in tangential bands between similar bands of tracheae or sieve-tubes.
The next change observable after some hours is that the untouched cells below the cut grow larger, push tip the dead surface, and divide by walls tangential to it, with the formation of tabloid cork-cells.
Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.
At a point whose distance from the axis of the wire is r the tangential magnetic force is H = 21r /a 2 (39) it therefore varies directly as the distance from the axis, where it is zero.'
A fluid is a substance which yields continually to the slightest tangential stress in its interior; that is, it can be divided very easily along any plane (given plenty of time if the fluid is viscous).
It follows that when the fluid has come to rest, the tangential stress in any plane in its interior must vanish, and the stress must be entirely normal to the plane.
Flow, Circulation, and Vortex Motion.-The line integral of the tangential velocity along a curve from one point to another, defined by s v as + u'a s) ds =f (udx+vdy-}-zdz), (I) is called the " flux " along the curve from the first to the second point; and if the curve closes in on itself the line integral round the curve is called the " circulation " in the curve.
The quiescent ellipsoidal surface, over which the motion is entirely tangential, is the one for which (a2+X)d?
This is called the tangential area, and will be denoted by T1.
The - - 4X + g tangential area may be expressed in terms of chordal areas.
Some of the formulae obtained by the above methods can be expressed more simply in terms of chordal or tangential areas taken in various ways.
There are similar formulae in terms of the tangential areas T1, Thus (iii) of § 68 may be written A -a -1-(9T 1 - T3).
The same method applies for tangential areas, by taking A -2 PTp -}- QT 4 + RT r + ..
Lord Rayleigh has shown that there is a tangential motion as well as a motion in and out.
Ordinarily when a bell is struck the impulse primarily excites the radial motion, and the tangential motion follows as a matter of course.
When a finger-glass (an inverted bell), is excited by passing the finger round the circumference, the tangential motion is primarily excited and the radial follows it.
Poynting may also be mentioned, in which the tangential component of the thrust of obliquely incident radiation is separately put in evidence, by the torsion produced in an arrangement which is not sensitive to the normal component or to the radiometer-pressure of the residual gas.
Comparing this equation with ux 2 +vy 2 +w2 2 +22G'y2+2v'zx+2W'xy=0, we obtain as the condition for the general equation of the second degree to represent a circle :- (v+w-2u')Ia 2 = (w +u -2v')/b2 = (u+v-2w')lc2 In tangential q, r) co-ordinates the inscribed circle has for its equation(s - a)qr+ (s - b)rp+ (s - c) pq = o, s being equal to 1(a +b +c); an alternative form is qr cot zA+rp cot ZB +pq cot2C =o; Tangential the centre is ap+bq+cr = o, or sinA +q sin B+rsinC =o.
Luminous arcs (T), tangential to the upper and lower parts of each halo, also occur, and in the case of the inner halo, the arcs may be prolonged to form a quasi-elliptic halo.1 The physical explanation of halos originated with Rene Descartes, who ascribed their formation to the presence of icecrystals in the atmosphere.
The "tangential arcs" (T) were explained by Young as being caused by the thin plates with their axes horizontal, refraction taking place through alternate faces.
Within the crust of the earth, whether by the contraction of the interior or in any other way, tangential pressures were set up. Since the crust is not of uniform strength throughout, only the weaker portions yielded to the pressure; and these were crumpled up against the more resisting portions and sometimes were pushed over them.
Secondary growth in thickness is effected by the tangential division of superficial cells.
The displacement of the point C of the body is made up of 10 tangential to the meridian ZC and sin 0 ~,1 perpendicular to the plane of this meridian.
Next suppose that the curve is rough; and let Fas be the tangential force of friction on s.
The tangential and normal components of was are ws sin ~ and --w&s cos, l.
The above problem is identical with that of the oscillation of a particle in a smooth spherical bowl, in the neighborhood of the lowest point, If the bowl has any other shape, the axes Ox, Oy may, ..--7 be taken tangential to the lines tof curvature ~ / at the lowest point 0; the equations of small A motion then are dix xdiy (II) c where P1, P2, are the principal radii of curvature at 0.
Mately, and the tangential acceleration at P is therefore dv/dt or 1.
Where p is the radius of curvature of the path at P, the tangential and normal accelerations are also expressed by v dv/ds and v1/p, respectively.
These expression~ therefore give the tangential and normal accelerations of P; cf.
At the instant t+t5t the momenta of the system are equivalent to a linear momentum represented by a localized vector ~(m).(U+U) in a line through G tangential to the path of G, together with a certain angular momentum.
Where K is the radius of gyration about the axis of symmetry, a is the constant distance of G from the plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig.
A curve tangential to all the sides of the polygon is the line of pressures.
And the component tangential to the joint is CQ=CRsin PCR=CPtan PCR.
If the joint be provided either with projections and recesses, such as murtises and tenons, or with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question of the utmost amount of the tangential resistance CQ which it is capable of exerting depends on the strength of such projections, recesses, or fastenings; and belongs to the subject of strength, and not to that of stability.
The condition of stability of friction is that the tangential component CQ of the resistance required shall not exceed the friction due to the normal component; that is, that CQ~f.CP,
Then Ob is the velocity of the point b in magnitude and direction, and cb is the tangential velocity of B relatively to C. Moreover, whatever be the actual magnitudes of the velocities, the instantaneous velocity ratio of the points C and B is given by the ratio Oc/Ob.
123 and 124) is the following: If points X and x are taken dividing the link BC and the tangential velocity cb, sothat cx: xb=CX:XB, then Ox represents the velocity of the point X in magnitude and direction.
The directions of the radial and tangential accelerations of the point B are always known when the position of the link is assigned, since these are to be drawn respectively parallel to and at right angles to the link itself.
If dw/dt is the angular acceleration of the link, dw/dt X CB is the tangential acceleration of the point B about the point C. Generally this tangential acceleration is unknown in magnitude, and it becomes part of the problem to find it.
The tangential pressures which are known to be set up in the earth's crust - either by the contraction of the interior or in some other way - caused the deposits of this sea to be crushed up against the rigid granites and other old rocks of the peninsula and finally led to the whole mass being pushed forward over the edge of the part which did not crumple.
The tangential polar equation to the epicycloid, as given above, is p= (a+2b) sin (a a+2b),I', while the intrinsic equation is s=4(bla)(a+b) cos (ala+2b)>G and the pedal equation is r2=a2+ (4b.a+b)p 2 l(a+2b) .
The megaspore-nucleus divides repeatedly, and cells are produced from the peripheral region inwards, which eventually fill the sporecavity with a homogeneous tissue (prothallus); some of the superficial cells at the micropylar end of the megaspore increase in size and divide by a tangential wall into two, an upper cell which gives rise to the short two-celled neck of the archegonium, and a lower cell which develops into a large egg-cell.
Araucaria, the leaf-traces persist for a considerable time, perhaps indefinitely, and may be seen in tangential sections of the wood of old stems. The leaf-trace in the Coniferales is simple in its course through the stem, differing in this respect from the double leaf-trace of Ginkgo.
Apart then from absorption there will be a discontinuous change in brightness in the apparent disk at that value of the angular radius d which corresponds to tangential emission from the upper lever r' of this mirage-forming region.