# U sentence example

From the equation K=(µ - I)/47r, it follows that the magnetic susceptibility of a vacuum (where µ = I) is o, that of a diamagnetic substance (where,

**u**I) is positive.Assume this true for

**u**1,**u**2, .,**u**,,.Then

**u**,,+1=un+(2n-+-I)=n2+(2n+I)=(n+I)2, so that it is true for**u**n+1.But it is true for

**u**1 .His travels and mercantile experience had led E t

**u**eopre him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both arithmetical and algebraical.AdvertisementIt vanishes when

**u**=mlr, m being any whole number other than zero.When

**u**= o, it takes the value unity.The maxima occur when

**u**=tan**u**, (4), and then sin 2**u**/**u**2 = cos 2**u**(5).To calculate the roots of (5) we may assume

**u**=(m+1)7r-y=**U**-y, (3), where y is a positive quantity which is small when**u**is large.It will readily be found that

**u**=**U**-y =**U**-**U**-1-**U**-a-15U 5 105U- -.AdvertisementSince the maxima occur when

**u**= (m +1)7r it nearly, the successive values are not very different from 4 4 4 &c The application of these results to (3) shows that the field is brightest at the centre =o, =0, viz.The part corresponding to negative values of

**u**is similar, OA being a line of symmetry.The obliquity, corresponding to

**u**=7, is such that the phases of the secondary waves range over a complete period, i.e.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.If we put for shortness 7 for the quantity under the last circular function in (I), the expressions (i), (2) may be put under the forms

**u**sin T, v sin (T - a) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin T and cos T in the expression**u**sin T +v sin (T - a), so that I =**u**2 +v 2 +2uv cos a, which becomes on putting for**u**, v, and a their values, and putting f =Q .AdvertisementComparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that G = z (cos

**u**+sin**u**)-M, H = z (cos**u**-sin**u**) +N.For example, when

**u**= o, M = o, N =o, and consequently G =H = 2.Descending series of the semi-convergent class, available for numerical calculation when

**u**is moderately large, can be obtained from (12) by writing x=uy, and expanding the denominator in powers of y.Peligot's results, though called in question by Berzelius, have been amply confirmed by all subsequent investigators; only now, on theoretical grounds, first set forth by Mendeleeff, we double Peligot's atomic weight, so that

**U**now signifies 240 parts of uranium, while UO 3 stands as the formula of the yellow oxide, and UO 2 as that of Berzelius's metal.Dilute sulphuric acid attacks it but slowly; hydrochloric acid, especially if strong, dissolves it readily, with the formation, more immediately, of a hyacinthcoloured solution of

**U**2 C1 6, which, however, readily absorbs oxygen from the air, with the formation of a green solution of UC1 4, which in its turn gradually passes into one of yellow uranyl salt, U02.AdvertisementDilute sulphuric acid precipitates uranium yellow, Na 2

**U**2 0 7.6H 2 O, from the solution so obtained.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.Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.-A stream-function 4, must be determined to satisfy the conditions v24 =o, throughout the liquid; (I) I =constant, over any fixed boundary; (2) d,t/ds = normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity

**U**in the direction Ox, d4y1ds=-Udy/ds, or 0 +Uy =constant over the moving cylinder; and 4,+Uy=41' is the stream function of the relative motion of the liquid past the cylinder, and similarly 4,-Vx for the component velocity V along Oy; and generally 1,1'= +Uy -Vx (4) is the relative stream-function, constant over a solid boundary moving with components**U**and V of velocity.Consider the motion given by w=

**U**(z+a2/z), (I) 4,=**U**(r+- r) cos 0=**U**+ a1x, so that (2) =**U**(r-)sin 0=**U**(i -¢) y.Over a concentric cylinder, external or internal, of radius r=b, 4,'=4,+ Uly =[

**U**I - + Ui]y, (4) and 4" is zero if**U**1 /**U**= (a 2 - b2)/b 2; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui; and w in (I) will give the liquid motion in the interspace between the fixed cylinder r =a and the concentric cylinder r=b, moving with velocity U1.AdvertisementIf the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = - Uz, we are left with w = Ua2/z, (6) =

**U**(a 2 /r) cos 0= Ua2x/(x2+y2), (7) 4, = -**U**(a 2 /r) sin 0= -Ua2y/( x2+y2), (8) giving the motion due to the passage of the cylinder r=a with velocity**U**through the origin 0 in the direction Ox.When the cylinder r =a is moved with velocity

**U**and r =b with velocity**U**1 along Ox, =**U**b e - a,1 r +0 cos 0 -**U**ib2 - 2 a, (r +Q 2 ') cos 0, = -**U**be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (**U**2 + V2)b2a a2U+Vi +b a b a2**U**z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (**U**b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (

**U**- Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (-**U**+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.An ellipse interior to n = a will move in a direction opposite to the exterior current; and when n = o,

**U**= oo, but V = (m/c) sh a sin 13.The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity

**U**reversed, and the cylinder is surrounded by a vortex.The components of velocity of the moving origin are denoted by

**U**, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if**u**, v, w denote the components of fluid velocity in space, and**u**', v', w' the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have**u**=**U**+**u**' - yR +zQ, v =V +v -zP +xR, w=W +w -xQ +yP.As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2

**U**¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then**u**= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.Thus with a' =o, a stream is split symmetrically by a wedge of angle ' zr/n as in Bobyleff's problem; and, by making a = oo, the wedge extends to infinity; then chnS2=

**u**, sh nS2= b n**u**.The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity

**U**and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity

**U**through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.Thus if T is expressed as a quadratic function of

**U**, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3,**U**= d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..In his valiant attempt to fill these gaps Rad„

**u**was obliged to invent kings and even dynasties, 13 the existence f which is now definitely disproved.Extraction of cane juice by diffusion (a process more fully described under the head of beetroot sugar manufacture) is adopted in a few plantations in Java and Cuba, in Louisiana Etr cti o n and the Hawaiian Islands, and in one or two factories y f i in Egypt; b

**u**t hitherto, except under exceptional conditions (as at Aska, in the Madras Presidency, where the local price for sugar is three or four times the London price), it would not seem to offer any substantial advantage over double or triple crushing.The mud collects at the bottom of the

**u**, and allows the upper part of the bag to filter for a longer time than would be the case if the bottom end were closed and if the bag hung straight like the letter I.In the same year appeared the first volume of the Hebrelisches

**u**.This in its simplest form gave rise to the rajaz verses, where each half-line ends in the same rhyme and consists of three feet of the measure -

**u**-.The violent tone of some of his printed manifestoes about this time, especially of his Lob des Konigs

**u**.Among swimming birds the most numerous are the gull (kamome), of which many varieties are found; the cormorant (

**u**)which is trained by the Japanese for fishing purposesand multitudinous flocks of wild-geese (gan) and wild-ducks (kanjo), from the beautiful mandarinduck (oshi-dori), emblem of cunjugal fidelity, to teal (koga,no) and widgeon (hidori-ganto) of several species.This arises from the pronunciation of

**u**as yu, and does not affect the English dialects which have not thus modified the**u**sound.The corresponding changes in the case of the mixture Tuvw are easily understood - the first halt at

**U**, due to the crystallization of pure B, will probably occur at a different temperature, but the second halt, due to the simultaneous crystallization of A and B, will always occur at the same temperature whatever the composition of the mixture.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.Then if

**U**is the potential outside the surface due to this electric charge inside alone, and V that due to the opposite charge it induces on the inside of the metal surface, we must have**U**+V =O or**U**= - V at all points outside the earthed metal surface.If we consider any short length of the stream bounded by two imaginary cross-sections A and B on either side of the plug, unit mass of the fluid in passing A has work, p'v', done on it by the fluid behind and carries its energy, E'+

**U**', with it into the space AB, where**U**' is the kinetic energy of flow.In passing B it does work, p"v", on the fluid in front, and carries its energy, E"+

**U**", with it out of the space AB.If there is no external loss or gain of heat through the walls of the pipe, and if the flow is steady, so that energy is not accumulating in the space AB, we must evidently have the condition E'+

**U**'+p'v' =E'+**U**"+p"v" at any two cross-sections of the stream.It is easy to arrange the experiment so that

**U**is small and nearly constant.In the limiting case of a long fine tube, the bore of which varies in such a manner that

**U**is constant, the state of the substance along a line of flow may be represented by the line of constant total heat, d(E+pv) = o; but in the case of a porous plug or small throttling aperture, the steps of the process cannot be followed, though the final state is the same.Scholz, Hubert Languet als kursachsischer Berichterstatter

**u**.None of them, in point of fact, has held its ground, and even his proposal to denote unknown quantities by the vowels A, E, I, 0,

**u**, Y - the consonants B, c, &c., being reserved for general known quantities - has not been taken up. In this denotation he followed, perhaps, some older contemporaries, as Ramus, who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, &c., only when these were exhausted.The os magnum 1, lunar; sc, scaphoid;

**u**, unciform; of the carpus articulates freely m, magnum; td, trapezoid; tm, with the scaphoid.Let

**u**represent the volume of air in the cup before the body was inserted, v the volume of the body, a the area of the horizontal FIG.The volume

**u**may be determined by repeating the experiment when only air is in the cup. In this case v =o, and the equation becomes (**u**--al l) (h - k') =uh, whence**u**= al' (h - k l) /k'.In one, applicable only to liquids which do not mix, the two liquids are poured into the limbs of a

**U**tube.The whole micrometer-box is moved by

**u**? ??'The number of molecules of the first kind of gas, whose components of velocity lie within the ranges between

**u**and**u**+du, v and v+dv, w and w+dw, will, by formula (5), be v?l (h 3 m 3 /7 3)e hm (u2+v2+w2)dudvdw (9) per unit volume.The cylinder is of volume

**u**dt dS, so that the product of this and expression (9) must give the number of impacts between the area dS and molecules of the kind under consideration within the interval dt.Thus the contribution to the total impulsive pressure exerted on the area dS in time dt from this cause is mu X udtdS X (11 3 m 3 /,r 3)e hm (u2+v2+w2 )dudvdw (I o) The total pressure exerted in bringing the centres of gravity of all the colliding molecules to rest normally to the boundary is obtained by first integrating this expression with respect to

**u**, v, w, the limits being all values for which collisions are possible (namely from - co too for**u**, and from - oo to + oo for v and w), and then summing for all kinds of molecules in the gas.The aggregate amount of these pressures is clearly the sum of the momenta, normal to the boundary, of all molecules which have left dS within a time dt, and this will be given by expression (pp), integrated with respect to

**u**from o to and with respect to v and w from - oo to +oo, and then summed for all kinds of molecules in the gas.Clearly the integral is the sum of the values of mu g for all the molecules of the first kind in unit volume, thus p=v mu l +v'm'

**u**2 +...Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and

**u**respectively.To illustrate the importance of the mensuration of graphs, suppose that we require the average value of

**u**with regard to x.The ordinate of the trapezette will be denoted by

**u**, and the abscissa of this ordinate, i.e.If there are m of these strips, and if the breadth of each is h, so that H =mh, it is convenient to write x in the form xo+Oh, and to denote it by x 0, the corresponding value of

**u**being ue.In the case of the briquette the position of the foot of the ordinate

**u**is expressed by co-ordinates x, y, referred to a pair of axes parallel to a pair of sides of the base of the briquette.In the same way the volume of a briquette between the planes x = xo, y = yo, x= a, y = b may be denoted by [[Vx,y ]y=yo]

**u**'x' =xo.The statement that the ordinate

**u**of a trapezette is a function of the abscissa x, or that**u**=f(x), must be distinguished from**u**=f(x) as the equation to the top of the trapezette.The simplest case is that in which

**u**is constant or is a linear function of x, i.e.The next case is that in which

**u**is a quadratic function of x, i.e.If we take these to be uo and

**u**2, and**u**1, so that m = 2, we have area = 6H(uo + 4u1 + u2) = 'h(uo + 4**u**1 + 142).If instead of uo,

**u**1, and**u**2, we have four ordinates uo, ul, u2, and**u**3, so that m = 3, it can be shown that area = 8h(uo + 3/41 + 3u2 - Fu3).Generally, if the area of a trapezette for which

**u**is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of**u**representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which**u**is an algebraical function of x of degree 2n + 1.When

**u**is of degree 4 or 5 in x, we require at least five ordinates.If m = 4, and the data are ul, u2, Us, U4, we have area = h (7

**u**o + 3 2u 2 -112/42 + 3 2u 3 + 7u4).Writing m = 2p, and grouping the coefficients of the successive differences, we shall find area = 2ph up+ 2 652up + 3 p4365p2 84up 3p,6 - 21p4 28p2 15120 If

**u**is of degree 2f or 2f + i in x, we require to go up to b 2f**u**p, so that m must be not less than 2f.The general formulae of § 54 (p being replaced (i) by 2m) may in the same way be applied to obtain formulae giving the area of the trapezette in terms of the mid-ordinates of the strips, the series being taken up to b 2f ul m or /th 2J ug m at least, where

**u**is of degree 2f or 2f + I in x.If

**u**is an algebraical function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q, may be expressed by a formula Aovo-1--yivi+..To extend these methods to a briquette, where the ordinate

**u**is an algebraical function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x = o is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y=o, through the given ordinates of the briquette (§ 44); and that the area of the section is then represented by the ordinate of a trapezette.Suppose, for instance, that

**u**is of degree not exceeding 3 in x, and of degree not exceeding 3 in y, that it contains terms in x3y3, x 3 y 2, x2y3, &c.; and suppose that the edges parallel to which x and y are measured are of lengths 2h and 3k, the briquette being divided into six elements by the plane x=xo+h and the planes y = yo+k, y = yo+2k, and that the 12 ordinates forming the edges of these six elements are given.The area of the section by a plane at distance x from the edge 0 is a function of x whose degree is the same as that of

**u**.The process is simplified by writing down the general formula first and then substituting the values of

**u**.The volume of the briquette for which

**u**is a function of x and y is found by the operation of double integration, consisting of two successive operations, one being with regard to x, and the other with regard to y; and these operations may (in the cases with which we are concerned) be performed in either order.In what follows it will be assumed that the conditions of continuity (which imply the continuity not only of

**u**but also of some of its differential coefficients) are satisfied, subject to the small errors in the values of**u**actually given; the limits of these errors being known.If the data are uo,

**u**1,.The justification of the above methods lies in certain properties of the series of successive differences of

**u**.The fundamental assumption is that each group of strips of the trapezette may be replaced by a figure for which differences of

**u**, above those of a certain order, vanish (§ 54).If we do not know values of

**u**outside the figure, we must use advancing or receding differences.The most simple case is that in which the trapezette tapers out in such a way that the curve forming its top has very close contact, at its extremities, with the base; in other words, the differential coefficients

**u**',**u**",**u**"',.These results may be extended to the calculation of an expression of the form fxo u4(x)dx, where 0(x) is a definite function of x, and the conditions with regard to

**u**are the same as in § 82.In cases other than those described in § 82, the pth moment with regard to the axis of

**u**is given by Pp = XPrA where A is the total area of the original trapezette, and S 2 _ 1 is the area of a trapezette whose ordinates at successive distances h, beginning and ending with the bounding ordinates, are o, x1P -1A, x2 P-1 (AI+AI),.The calculation of the expressions in brackets may be simplified by taking the pairs in terms from the outside; by finding the successive differences of uo + um, ill + um_l, ..., or of uI

**u**i +umi, ..Parinentier's rule, for instance, assumes that in addition to

**u**I;**u**I..If the data of the briquette are, as in § 86, the volumes of the minor briquettes, but the condition as to close contact is not satisfied, we have y "`x P

**u**dx dy = K + L + R - X111010-0,0 f xo yo i'?Either or both of the expressions K and L will have to be calculated by means of the formula of § 84; if this is applied to both expressions, we have a formula which may be written in a more general form f f 4 u4(x, y) dx dy =

**u**dx dy.The Euler-Maclaurin formula (§ 75) assumes that the bounding values of

**u**',**u**"',..It is also clearly impossible to express

**u**as an algebraical function of x and y if some value of du/dx or duldy is to be infinite.One method is to construct a table for interpolation of x in terms of

**u**, and from this table to calculate values of x corresponding to values of**u**, proceeding by equal intervals; a quadrature-formula can then be applied.Even where

**u**is an explicit function of x, so that f x udx may be expressed in terms of x, it is often more convenient, for construction of a table of values of such an integral, to use finite-difference formulae.Attention must be given to the possible accumulation of errors due to the small errors in the values of

**u**.The greater of the two temples was sacred to Jupiter (Baal), identified with the Sun, with whom were associated Venus and Mercury as a-p,

**u**co,uoc Beni.If each wave travels out from the source with velocity

**U**the n waves emitted in one second must occupy a length**U**and therefore**U**= nX.Generally, if any condition in the wave is carried forward unchanged with velocity

**U**, the change of 4 at a given point in time dt is equal to the change of as we go back along the curve a distance dx = Udt at the beginning of dt.But it has velocity

**U**, and therefore momentum poU 2 is carried in.If the velocity of a particle at A relative to the undisturbed parts is

**u**from left to right, the velocity of the matter moving out at A is**U**-**u**, and the momentum carried out by the moving matter is p(**U**-**u**) 2.There is also the " external " applied pressure X, and the total momentum flowing out per second is X-I-P4-W-1-p(

**U**-**u**)2.Equating this to the momentum entering at B and subtracting P' from each X+W+p(

**U**-**u**)2 =poU 2.If then we apply a pressure X given by (5) at every point, and move the medium with any uniform velocity

**U**, the disturbance remains fixed in space.Or if we now keep the undisturbed parts of the medium fixed, the disturbance travels on with velocity

**U**if we apply the pressure X at every point of the disturbance.If the velocity

**U**is so chosen that E - poU 2 = o, then X = o, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity**U**= (E/p).If, however, we put on external forces of the required type X it is obvious that any wave can be propagated with any velocity, and our investigation shows that when

**U**has the value in (6) then and only then X is zero everywhere, and the wave will be propagated with that velocity when once set going.He supposed that in air Boyle's law holds in the extensions and compressions, or that p = kp, whence dp/dp = k = p/p. His value of the velocity in air is therefore

**U**= iJ (p ip.) (Newton's formula).That is to say, instead of using Boyle's law, which supposes that the pressure changes so exceedingly slowly that conduction keeps the temperature constant, we must use the adiabatic relation p = kpy, whence d p /d p = y k p Y 1= yp/p, and

**U**= (yp/p) [Laplace's formula].In the momentum equation (4) we may now omit X and it becomes 0.+P(

**U**-**u**) 2 =poU2.We have

**U**-**u**=**U**(I -**u**/**U**) =**U**(1 - v/V), since**u**/**U**= - dy/dx= v/V.But for very small times the assumption may perhaps be made, and the result at least shows the way in which the velocity is affected by the addition of a small term depending on and changing sign with

**u**.We see at once that, where

**u**=o, the velocity has its " normal " value, while where**u**is positive the velocity is in excess, and where**u**is negative the velocity is in defect of the normal value.Meanwhile the waves are spreading out and the value of

**u**is falling in inverse proportion to the distance from the source, so that very soon its effect must become negligible.This is hardly to be explained by equation (I I), for at the very front of the disturbance

**u**=o and the velocity should be normal.The kinetic energy per cubic centimetre is 2 pu t, where is the density and

**u**is the velocity of disturbance due to the passage of the wave.But the values of 2 which occur successively during the second at AB exist simultaneously at the beginning of the second over the distance

**U**behind AB.Or if the conditions along this distance

**U**could be maintained constant, and we could travel back along it uniformly in one second, we should meet all the conditions actually arriving at AB and at the same intervals.We also have pu t =p o

**u**t /(I +dy/dx).We have

**U**-Fw= D/T 1 and**U**- w = D/T .Comparing the velocities of sound

**U**i and U2 in two different gases with densities and at the same temperature and pressure, and with ratios of specific heats 'yl, 72, theory gives Ui/U2 = 1/ {71 p 2/72 p i }.Since

**U**=n X where**U**is the velocity of sound, X the wave-length, and n the frequency, it follows that the forward frequency is greater than the backward frequency.Let S' be its position one second later, its velocity being

**u**.Let the velocity of the air from S to R be w, and let

**U**be the velocity of sound in still air.Then the n waves occupy a space

**U**+ w -**u**.If he were at rest, it would be the waves in length

**U**+ w, for the wave passing him at the beginning of a second would be so far distant at the end of the second.But through his motion v in the second, he receives only the waves in distance

**U**+ w - v.Since there are n waves in distance

**U**+ w -**u**the number he actually receives is n(**U**+ w - v)/(**U**+ w -**u**).If now the curve moves along unchanged in form in the direction ABC with uniform velocity

**U**, the epoch e =OA at any time t will be Ut, so that the value of y may be represented as 2 y=a sin T (x - Ut).It is therefore W m = (E 1

**U**) 27rUa/A = 27rnpUa.It is, therefore,

**u**m.If we measure t from an instant at which the two trains exactly coincide, then as

**U**for the other train has the opposite sign, its displacement is represented by y2= a sin (x+Ut).If

**U**4 is the velocity of longitudinal waves along the sounder, and 1 the length of the sounder, the frequency of vibration is**U**8 /2l.If L 1 is the internodal distance and

**U**1 the velocity in a gas, L and**U**being the corresponding values for air, we have**U**1 /**U**=L1/L.If

**U**is the velocity of sound in a gas at pressure P with density p, and if waves of length X and frequency N are propagated through it, then the distanc?e l between the dust-heaps is 2 = N - zN Vyp' where y is the ratio of the two specific heats.Suppose that a disturbance is travelling with velocity

**U**unchanged in form along a rod from left to right.At B there is only the latter kind, and since the transfer of matter is powoU, where po is the undisturbed density and wo is the undisturbed cross-section, since its velocity is

**U**the passage of momentum per second is powoUo 2.At A, if the velocity of the disturbance relative to undisturbed parts of the rod is

**u**from left to right, the velocity relative to A is**U**-**u**.If p is the density at A, and w the cross-section, then the momentum carried past A is pc,(

**U**-**u**) 2.Then we have Ya,ody/dx +p& (

**U**-**u**) 2 = p o&,oU 2.The velocity with which the rod must travel in order that the disturbance may be fixed in space is therefore

**U**=, I (Y/p), or, if the rod is kept fixed, this is the velocity with which the disturbance travels.The mass of matter moving through A per second is pwa 2

**U**, where a is the radius of the wire and p is its density.Substituting in (33) we get

**U**2 = n/p. (34) If we now keep the wire at rest the disturbance travels along it with velocity**U**= d (nip), and it depends on the rigidity and density of the wire and not upon its radius.He also published various treatises of archaeological interest, of which the most important are Die Erfindung des Alphabets (1840), Urgeschichte

**u**.Deep and narrow canons are common, and, at higher levels, glaciers, carved out amphitheatres, or " cirques " and "

**U**"-shaped troughs.Let t be the statical breaking strength of a bar, loaded once gradually up to fracture (t = breaking load divided by original area of section);

**u**the breaking strength of a bar loaded and unloaded an indefinitely great number of times, the stress varying from**u**to o alternately (this is termed the primitive strength); and, lastly, let s be the breaking strength of a bar subjected to an indefinitely great number of repetitions of stresses equal and opposite in sign (tension and thrust), so that the stress ranges alternately from s to -s.Wohler's and Bauschinger's experiments give values of t,

**u**, and s, for some materials.We shall make the natural supposition that motion of the aether, say with velocity (

**u**,v,w) at the point (x,y,z), is simply superposed on the velocity V of the optical undulations through that medium, the latter not being intrinsically altered.But in the transition from molecular theory to the electrodynamics of extended media, all magnetism has to be replaced by a distribution of current; the latter being now specified by volume as well as by flow so that (

**u**,v,w) ST is the current in the element of volume 6T.His first work was Der Montanismus

**u**.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.Also the velocity v at the end of the arc is given by (87) ve =

**u**e sec 0 cos n.Tupinambis teguixin, the "tej

**u**" of South America and the West Indies, is the largest member of the family; it reaches a length of a yard, most of which, however, belongs to the strong, whip-like tail.Oppert (24) concludes from inscriptions that there was in Assyria a royal cubit (7/6)ths of the

**U**cubit, or 25.20; and four monuments show (25) a cubit averaging 25.28.It is described by Oppert (24), from literary sources, as the great

**U**of 222 susi or 39.96, double of 19.98; from which was formed a reed of 4 great**U**or 159.8.Among his numerous literary works are Uber Pclychromie

**u**.The figures of the Seven have here entirely disappeared, the remembrance of them being merely preserved in the name of the Agpaovp y y6s (050,

**u**s).Jocham, Geschichte des Lebens

**u**.An interesting little work is Klaiber, Holderlin, Hegel,

**u**.Its length (inclusive of the flagellum) varies from 40-60 while its greatest width (including the undulating-membrane) is from 8-30,

**u**; in the very wide individuals breadth is gained more or less at the expense of length.The "house for bloodletting and purging" adjoins it on the west (

**U**).The entrance was by a strong gateway (

**U**) to the north.Parallel with the western walk is an immense vaulted substructure (

**U**), incorrectly styled the cloisters, serving as cellars and store-rooms, and supporting the dormitory of the conversi above.This vast area, shaped like a broad-limbed V or

**U**, with Hudson Bay in the centre, is made up chiefly of monotonous and barren Laurentian gneiss and granite; but scattered through it are important stretches of Keewatin and Huronian rocks intricately folded as synclines in the gneiss, as suggested earlier, the bases of ancient mountain ranges.It will be seen that the umbilical fissure (

**u**) divides the organ into right and left halves, as in the lower vertebrates, but that the ventral part of each half is divided into a central and lateral lobe.The valves of the shell have been removed by severing their adhesions to the muscular areae h, i, k, 1, m,

**u**.Thus, to take an example, he will not print a critical text of Plautus with two letters (Y and Z) which were no part of the Latin alphabet in the age of that comedian; still less will he introduce into Latin texts distinctions, such as i,j and

**u**, v, which were not used till long after the middle ages.A distinction (chap. 2) is drawn between things which are predicates of a subject (Kae'

**U**?oKEi b tevov) and things which inhere in a subject (iv U7roKEL ivC J); and, while universals are called predicates of a subject, things in a subordinate category, i.e.It divides (chap. 8) evidences (7r1aTEts) into two kinds (I) evidence from arguments, actions and men (ai j s v E air&v Tcev Xhywv cal Twv 7rpit aw cal TWV avOpcoirwv); (2) adventitious evidences (ai S' iIriOETOtTOLs X yo,

**u**vots cal Tois rpm-To/lb/0a).This court, which the American government proposed to call a " Court of Arbitral Justice," would take the place of that which it was proposed to institute under Vc

**u**No.The intention of the Hague draft annexed to the Vc

**u**was to create a permanent court as distinguished from that established in 1899, which, though called permanent, was not so, having to be put together ad hoc as the occasion arose.In the case of hydrogen rendered luminous in a vacuum tube we may put approximately

**u**equal to 2000 metres per second, if the translatory motion of the luminous molecules is about the same as that at the ordinary temperature.The distance between the lines measured on the frequency scale does not, according to the equation, increase indefinitely from the head downwards, but has a maximum which, in Pickering's form as written above, is reached when (s +,

**u**) 2 = 3a.We wait, however, for the Epistles of his captivity at Rome to find the full meaning of the idea of the church dawning

**u**p on his imagination.The vowel sounds ai, oi, ui have become e, o,

**u**; and a, o,**u**before the finals d and n are now et, o, ii.The first and fourth became masters of Tsangrong, the second, took possession of Amdo and Tsongkha, the third became king of

**U**(or the central Lhassan province), and removed the capital to Yarlung, south of Lhasa.While the dynasty of Khorre in Shang-shung and that of Thich'ung in

**U**were running, another authority, destined to become the superior of both, had arisen in Tibet.The kings of

**U**greatly patronized them, as for instance in the case of the celebrated Sakya Pandita by the seventh of these kings.In return for his services, Kublai invested Phagspa with sovereign power over (1) Tibet proper, comprising the thirteen districts of

**U**and Tsang, (2) Khalil and (3) Amdo.If l is the heat of dilution per unit change of volume in a calorimeter where all the energy goes to heat, the change in internal energy

**U**is measured by ldv.Now the velocities

**u**and v of the opposite ions under unit potential gradient, and therefore**U**and V under unit force, are known from electrical data.About fifty species of Saccharomyces are described more or less completely, but since many of these cannot be distinguished by the microscope, and some have been found to develop physiological races or varieties under special conditions of - ?

**u**growth, the limits are still far too ill-defined for complete ep botanical treatment of the genus.Eutectic here freezes 7 Austenite+Cementite Pro-eutectoid Cementite forms progressiuely 882930 1200 0

**U**a 5 6 ?**U**, Key-link holding bottom.In short, there can be no doubt that the biblical name Shinar was practically equivalent to the mat Shumeri

**u**Akkadi= non-Semitic Kengi-Uri of the Babylonian inscriptions.For example, an indeterminative vowel, a, e, i or

**u**, may be prefixed to any root to form an abstract; thus, from me, " speak," we get e-me, " speech"; from ra, " to go," we get a-ra, " the act of going," &c. In connexion with the very complicated Sumerian verbal system 2 it will be sufficient to note here the practice of infixing the verbal object which is, of course, absolutely alien to Semitic. This phenomenon appears also in Basque and in many North American languages.In Germany it is sometimes designated Stiller Freitag (compare Greek, Ef3bo,

**u**&s t17rpaKro; Latin, hebdomas inoficiosa, non laboriosa), but more commonly Charfreitag.After the fall of Napoleon and the conclusion of the first peace of Paris (30th of May 1814) Belgium was indeed for some months placed under the administration of an Austrian governor-general, but it

**u**f was shortly afterwards united with Holland to form the kingdom of the Netherlands.See Liebscher, Charron

**u**.The element a means "water," and in

**u**-sar it is probable that**u**also means "water," while sar is "park, district."The limbs of the

**U**are further twisted together in a looser or tighter coil, the axis of which may be traversed by a "spindle" muscle arising from the posterior end of the body.Trenton is served by the Pennsylvania (main line and Belvidere division) and the Philadelphia & Reading railway systems, by inter-urban electric railways, and by small freight and passenger steamers on the Delaware river; the Delaware && Raritan Canal connects with r 0

**U**Argent Diagram of Half of Trente et Quarante Table.**U**, gradually became obsolete), plural (m.Having begged for quarter First and surrendered, they were immediately stripped nearly massacre naked, and about fifty were slaughtered on the spot; of the and about the same number were dragged away, with Mame- every brutal aggravation of their pitiful condition, to

**U**es.To these troops their chief now made known the pashas orders to massacre all the Mamelukes within the citadel; therefore, having returned Final by another way, they gained the summits of the walls massacre and houses that hem in the road in which the Mameof the lukes were confined, and some stationed themselves Manic- upon the eminences of the rock through which that

**U**es.Kolbing is printed in the Jahrbuch fiir romanische

**u**.Driver, Modern Research as illustrating the Bible (1909); P. Thomsen, Paldstina

**u**.Strabo follows up the topographical data with a few brief historical statements - "OaKot €t ov Kai raur'v Kai 111v e0-js no,

**u**?rniav.Vearnig /islands Continuation North Same Scale k, Ningayen_ CD tie Leyte Dinagat Siargao 6 14 0

**u**/ ........................We are told that an assembly was stirred to wild applause by a double trochee - -

**u**.Evaporation of a solution at ordinary temperatures gives colourless monoclinic prisms of Th(SO 4) 2.9H 2 O, which is isomorphous with uranium sulphate,

**U**(S04)2.9H20.Combebiac does not use K; and in place of, n he uses,

**u**=1 7 - E, so that,**u**2 = I, wµ = - µw = w, w 2 = o.When Su = o, (I + 2wu) () (1 + zwu) -1 is an operator which shifts (without further change) the tri - quaternion operand an amount given by

**u**in direction and distance.This page gives an overview of all articles in the 1911 Brittanica which are alphabetized under

**U**to Uzz.In the " cockney " dialect, really the dialect of Essex but now no less familiar in Cambridge and Middlesex, the ai sound of i is represented by of as in toime, " time," while a has become ai in Kate, pane, &c. In all southern English o becomes more rounded while it is being pronounced, so that it ends with a slight

**u**'sound.At a much later epoch it was introduced into the Latin alphabet by the emperor Claudius to represent y, and the sound which was written as i or

**u**in maximus, maxumus, &c.These can again be combined into a single resultant R acting in the direction (X,

**u**, v), provided X=RX,YRu,ZRv.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.It is easily inferred as a limiting case, or proved directly, that two infini tesimal rotations a, j3 about c

**u**parallel axes are equivalent to a ..._ -- - -The displacement will consist of an infinitesimal rotation e about some axis through

**U**, whose direction-cosines are, say, 1, m, n.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.We assume that in limiting equilibrium we have F tsR, everywhere, where

**u**is the coefficient of friction.A curve with I as abscissa and

**u**as ordinate is called the curve of velocities or velocity-time curve.This applies to the inverted pendulum, with

**u**=g/l, but the equation (9) is then only approximate, and the solution therefore only serves to represent the initial stages of a motion in the neighborhood of the position of unstable equilibrium.Again, the time of falling from a distance a into a given centre of force varying inversely as the square of the distance will depend only on a and on the constant

**u**of equation (15).The argument appears in a more demonstrative form in the theory of similar systems, or (more precisely) of the similar motion of similar systems. Thus, considering the equations d2x

**u**dix u1Obviously 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.Again, in the equiangular spiral we have p =r sin a, and therefore P =

**u**/ri, if**u**=hh/sinh a.The law of the inverse cube P=

**u****u**is interesting by way of contrast.The equation of the latter, referred to its principal axes, being as in II (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or X,

**u**, v.The motion of a rigid body in the most general case may be specified by means of the component velocities

**u**, v, w of any point 0 of it which is taken as base, and the component angular velocities p, q, r.The components of angular momentum about Ox, Qy, 01 will be denoted as usual by X,

**u**, v.If we now apply them to the case of a rigid body moving about a fixed point 0, and make Ox, Oy, Oz coincide with the principal axes of inertia at 0, we have X,

**u**, v=Ap, Bq, Cr, whence A (B C) qr = L,When, in any problem, the values of

**u**, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference.The meaning of these quantities is easily recognized; thus X is the angular momentum about a horizontal axis normal to the plane of 0,

**u**is the angular momentum about the vertical OZ, and s is the angular momentum about the axis of symmetry..If M be large compared with in,

**u**is small, and the roots are g/a and g/b, approximately.The efficiency corresponding to the last equation is

**U**IThe sectarial mark of the Ramanujas resembles a capital

**U**(or, in the case of another division, a Y), painted with a white clay called gopichandana, between the hair and the root of the nose, with a red or yellow vertical stroke (representing the female element) between the two white lines.Their sectarial mark is like the

**U**of the SriVaishnavas, except that their central line is black instead of red or yellow.In Latin it is regularly used in combination with

**u**.In classical Latin its use is confined to the cases where, as in English quill, &c., the

**u**is pronounced as w before a following vowel, but in old Latin it is found also in other combinations.Many languages find the combination qu, when both sounds are consonantal (qw), difficult; q being the deepest guttural while

**u**(English w) is a lip sound, the points of production are nearly as far separate as they can be.Welsh had the sound of French

**u**, but now has the clear sound of y described above, which is similar to the ear, and has the same pitch.Aryan a, i,

**u**remained.Probably also Celtic

**u**was advancing or had advanced to a forward position, for it appears in Welsh as I, as in din, " stronghold," from Celtic *dun-on, cognate with Eng.Latin o and

**u**appear as**u**; see examples above.If S (fig.5) be any optical system, rays proceeding from an axis point 0 under an angle

**u**l will unite in the axis point O'1; and those under an angle 24 2 in the axis point 0'2.If there be refraction at a collective spherical surface, or through a thin positive lens, 0' 2 will lie in front of O' 1 so long as the angle u2 is greater than

**u**1 (" under correction "); and conversely with a dispersive surface or lenses (" over correction ").This distance replaces the angle

**u**in the preceding considerations; and the aperture, i.e.If the object point 0 be infinitely distant,

**u**, and**u**2 are to be replaced by h l and h 2, the perpendicular heights of incidence; the " sine condition " then becomes sin**u**',/h 1 = sin**u**' 2 /h 2 .If we form then the continued fraction inwhich pi, p2, p3 9 ..., pn are u1, u1 +

**u**2, ul+u2+u3,The second minus the first, or the increase in the sum of tensions, is thus 2 (

**U**n - o' n+l) (o n+ 1 - Q n +2) Hence, if an+1 be intermediate in magnitude between a,, and a71+2, the sum of tensions is increased by the abolition of the stratum; but, if a-n+1 be not intermediate, the sum is decreased.We have, therefore, to consider the conditions under which n2172 (V a +5) T(p -

**U**)g cannot be made negative.H -A-R B_O

**U**Walker & ockerell that formed the chief entrance of the precinct of Apollo.The flame-like P

**u**t the matter in another processes and outliers are composed of way, if we could imagine writhing filaments, and the contours all the living cells of a are continually changing while the large oak or of a horse, colony moves as a whole.For a full description of the cathedral, in Serbo-Croatian and French, see the finely illustrated folio Stolna Crkva

**u**Djakovu, published by the South Slavonic Academy (Agram, 1900).The vowels a, e, i, (y), o,

**u**, are pronounced as in Italian; but e = Eng.His upon system is based

**u**on the formal denial of a trans cendent deity.V?'o?took?s,l eSk ham o h `

**u**Br1 n NewC l 4 ',fteg, r,a, t Sha?Luku!

**u**A Longitude E.ust of Capitol Pygmy Forest (from the races inhabiting it), the Aruwimi or Ituri Forest (from the rivers traversing it), the Stanley Forest (from its discoverer), or the Great Congo Forest.It is represented by a number of local races, mostly of smaller size, such as the Burmese and Malay C.

**u**.Thus, at Delphi there was an image of Aphrodite 6rtrupt31a (" Aphrodite of the tomb "), to which the dead were summoned to receive libations; the epithets ru,

**u**i 3capvxos (" grave-digger "), µvxia (" goddess of the depths "), peXacv%s (" the dark one "), the grave of Ariadne-Aphrodite at Amathus, and the myth of Adonis, point in the same direction.At this Ch rch moment they were more prevalent than ever, largely

**U**in consequence of the way in which the popes at Avignon had made themselves the allies and tools of the kings of France.The moment at which he made this

**U**,ird third attempt was one of unusual anxiety.Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of

**u**in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).It is obvious that we cannot by consideration of the equation

**u**= o in point-co-ordinates obtain the remaining three of Pliicker's equations; they might be obtained in a precisely analogous manner by means of the equation v= o in line-co-ordinates,but they follow at once from the principle of duality, viz.The Hessian A has just been spoken of as a covariant of the form

**u**; the notion of invariants and covariants belongs rather to the form**u**than to the curve**u**=o represented by means of this form; and the theory may be very briefly referred to.If, however, the geometrical property requires two or more relations between the coefficients, say A = o, B = o,&c., then we must have between the new coefficients the like relations, A' = o, B' = o, &c., and the two systems of equations must each of them imply the other; when this is so, the system of equations, A = o, B = o, &c., is said to be invariantive, but it does not follow that A, B, &c., are of necessity invariants of

**u**.Similarly, if we have a curve

**U**= o derived from the curve**u**= o in a manner independent of the particular axes of co-ordinates, then from the transformed equation**u**' = o deriving in like manner the curve**U**' = o, the two equations**U**= o,**U**' = o must each of them imply the other; and when this is so,**U**will be a covariant of**u**.The case is less frequent, but it may arise, that there are covariant systems

**U**= o, V = o, &c., and**U**' = o, V' = o, &c., each implying the other, but where the functions**U**, V, &c., are not of necessity covariants of**u**.The theory of the invariants and covariants of a ternary cubic function

**u**has been studied in detail, and brought into connexion with the cubic curve**u**= o; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.He is known as the author of monographs upon Georg Calixt

**u**.See C. Hegel, Stadte

**u**.Basing his conclusions upon philological data, such as the names of wheat in the oldest known languages, the writings of the most ancient historians, and the observations of botanical travellers, De Candolle infers that the hdistromeib

**u**a n d original home of the wheat plant was in Mesopotamia, don.From S t r a s b

**u**r g e is Lehrbuch der Botanik, by permission of Gustav Fischer.In order to represent in the figure the 2 position of the f

**u**ndamental plane, we conceive a circle to be drawn round 0, lying in that plane.Let us now consider an equation of the form

**U**=a sin (nt+Lo).The value of

**U**will continually oscillate between the extreme values +a and - a, going through a series of changes in the same period in which the angle nt+Lo goes through a revolution.In this case the motion of

**U**, while still periodic, is seemingly irregular, being much like that of a pitching ship, which has no one unvarying period.In western New Guinea, according to the Dutch missionaries, there is a vague notion of a universal spirit, practically represented Spirit by several malevolent powers, as Manoin, the mostn the woods; Narw, in the worship. c p louds,

**u**above the trrees, l a sort of Erl-Konig h o carries off children; Faknik, in the rocks by the sea, who raises storms. As a protection against these the people construct - having first with much ceremony chosen a tree for the purpose - certain rude images called karwars, each representing a recently dead progenitor, whose spirit is then invoked to occupy the image and protect them against their enemies and give success to their undertakings.He was the syncellus (cell-mate, the confidential campanion assigned to the patriarchs, sometimes little more than a spy; see Syncellus) or private secretary of Tara(

**u**)sius, patriarch of Constantinople (784-806), after whose death he retired to a convent, and wrote his Chronicle of events from Adam to Diocletian (285).The act of 1870, ad n

**u**ttin the divergence between theory andpractice, League.It is clear that direct transmission through the plate at a point where the thicknesses of the prisms are d 1 and d 2 will introduce a relative retardation of (µ,; -,

**u**o) (d l - d2) between streams polarized in planes parallel and perpendicular to the edges of the prisms,,**u**o, and being the ordinary and the extraordinary refractive indices; and it is hence possible by an adjustment of the thickness to reduce elliptically polarized to plane polarization at an assigned point marked off by two parallel lines.The Latin never yields ie in Catalan as it does in French and occasionally in Provenal; s e d e t becomes seu (where

**u**represents the final d), p e d e m makes peu, and e go eu; in some words where the tonic is followed by a syllable in which an i occurs, it may become I (ir, he r i; mig, me di us; m-,is, m eli us); and the same holds good for in a similar situation (ciri, c r i**u**s, c e r e**u**s; fire, f e r i a), and for e in a close syllable before a nasal (eximpli, e x e to p 1**u**m; mintr for mentir, gint for gent).I tonic long and i short, when in hiatus with another vowel, produce i (amich, a in i c

**u**1; via, vi a).Just as e before a syllable in ivhich an i occurs is changed into I, so in the same circumstances o becomes

**u**(full, folium;vuil, volio forvoleo)andalsowhentheaccented vowel precedes a group of consonants like ci, p1, and the like (ull, o c 1**u**s; escull, Sd 0 p I**u**s).Latin

**u**persists with the Latin pronunciation, and, as already said, does not take the FrancoProvenal pronunciation it.In the same way the supporting vowel, which is regularly an e in CataIan, is often written a, especially after r (abra, ar bore m; astra, a s t r

**u**m; para, p a t r e in); one may say that in the actual state of the language post-tonic e and a become indistinguishable in a surd sound intermediate between the French a and mute e.Bet ore the tonic the same change between a and e constantly takes place; one finds in manuscripts enar, emor for anar, amor (the same extends even to the case of the tonic syllable, ten and sent from t a n t

**u**in and 1 a n c t**u**m being far from rare), and, on the other hand, antre, arrar, for entre, errar.I atonic is often represented by e even when it is long (vehi, v i c i n

**u**s).**U**atonic keeps its ground.The only strong diphthongs of the spoken language are di, du (rather rare), ei,

**u**-, -ia, oi,**u**, di, uu.Final d after a vowel has produced

**u**(pea, p e d e in; niu, n i d**u**m; mou, to o d**u**m); buf when the d, in consequence of the disappearance of the preceding vowel, rests upon a consonant, it remains and passes into the corresponding surd; f r I g i d**u**s gives fred (pronounced fret).Final n, if originally it stood between two vowels, drops away (bo, b 0 n

**u**m; vi, v i n**u**m), but not when it answers to mn (thus do nu in makes do, but dom num don; sonum makes so, but somnum son).B is replaced by the surd pat the end of a word (trobar in the infinitive, but trop in the present tense); so also in the interiOr of a word when it precedes a consonant (supvensr, s

**u**b v e n i re, sopte, s**u**b t 0).Median intervocalic f gives v (Estve, S t ep h an

**u**s); it has disappeared from p r 0 f**u**n d**u**s, which yielded the form preon, then pregon (g being introduced to obviate the,hiatus).V, wherever it has been preserved, has the same pronunciation as at the end of a word and between vowels it becomes vocalized into

**u**(suau, s**u**a vi s; viure, vi v e r e).Lj and

**U**give I mouillc (11 in the present orthography - fill, e i Ii**u**m; consell, co nsili urn; null, n ullu in).Rs is reduced to s (cos for cons, c 0 r p

**u**s).In this feature, and in its almost universal conservation of the final vowels e, i,

**u**(o), Castiian comes very near Italian, while it separates from it and approaches the Gallo-Roman by its modification of the consonants.Quien, the interrogative pronoun which has taken the place of the old qul, seems to come from q

**u**e rn.The Latin diphthong an is rendered in Portuguese by ou (euro, a

**u**r**u**mu; pouco, p a**u**c ii m), also pronounced of.It is stated, for example, that Gallego does not possess nasal diphthongs; still it may be conceded once for all that such a word as p 1 a n

**u**s, which in Galician is written sometimes chau and sometimes c/ian, cannot be very remote from the Portuguese nasal pronunciation chao.If the value of the clearness in air be taken as sin u1, then by the law of refraction N =sin

**u**l /sin u2, the value for the clearness in water is N sin**u**2.The function n sin

**u**= A, for the microscope, has been called by Abbe the numerical aperture.On the other hand, in immersion-systems the numerical aperture can almost amount to the refractive index, for A=n sin

**u**The value n sin

**u**equals the numerical aperture A, where n is the refractive index of the immersion-liquid, and**u**is the semi-aperture on the object-side.With sunlight particles can be made visible to a size of about o o04,

**u**.The sine-condition is in contrast to the tangent-condition, which must be regarded as the point-by-point representation of the whole object-space in the image-space (see Lens), and according therefore the equation n tan

**u**/tan**u**' = C must exist.The upper lip

**u**is composed of two petals united, the lower lip (1) of three.The former was called simply o and the latter

**u**(ov, pronounced as oo in moon).Though short o changed in the Latin of the last age of the Roman republic to

**u**in unaccented syllables always (except after**u**whether vowel or consonant), and sometimes also in accented syllables, this was not equally true of vulgar Latin, as is shown by the Romance languages.U1 -

**U**', leaf-traces, numbered according to the phyllotaxis, It 5 belonging to the lowest leaf of the five; ph, a group of primary phloem; pd, periderm, formed from pericycle.Bro, ur supposed 2 tell me when

**u**visit.**U**still online? he typed.After some work, they found there were nineteen different letters used, with "E, I, L, O,

**U**, Q, and W" missing.In this study, unlike the Saskatchewan study, urine albumin was not related to

**U**intake.In the middle of the '

**U**', the canal crosses an aqueduct over the river.If

**u**get one done make it a nice one... wat do**u**ave in mind?Lire " how can

**u**say ciara found bowwow in a hotel with a stripper?Are you honestly trying to draw a comparison with Man

**U**?Hydrogen atoms for which 3D coordinates are available are taken into account and assigned

**U**iso values of 0.06 Ã… 2.**U**films should be set within a positive moral framework and should offer reassuring counterbalances to any violence, threat or horror.Get someone to use it on

**u**coz its a slippery bugger lol How would you rate this item out of 5?Hope

**u**enjoyed it Can i jus say dat ive dun dis 4 half a year!I'm writing

**u**dis bcoz I'm hoping on hope dat**u**'ll love life and live it 2 d fullest.We know from previous actions of this administration what B

**u**sh, Ch en ey and Rum sf eld are capable of.The principal axes of the thermal ellipsoid can be obtained from the

**U**values via a principal axes transformation.If

**u**Eva need 2 tlk 2 sum1 i'm hear 2 lisen - sorry bout ure dad Luv Nicky xXx.I think

**u**will Neva Eva eva will gt ova a loss of sum1!The three Gemmell sisters have proved their seeding and are all in the semi finals today, Girls

**U**13, 15, 17.Once a

**U**shaped hole has been formed a fissure is created between two dunes.Click here to return to the present Guestbook hey guys, saw

**u**in the newspaper - stockport times!I notice

**u**guys live in USA - I am uk.But used corectly it can be a very good thing to use. y are

**u**thinking of using a hackamore?That's right, the Duo has an RJ-45 jack for the integrated NT-1 ISDN

**U**interface, plus an RJ-11 telephone jack.