U sentence example

u
  • He immediately turned on his lights and made a quick U turn.
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  • Baentsch draws attention to this feature in his monograph Altorientalischer u.
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  • From the ovo-testis, which lies near the apex of the visceral coil, a common hermaphrodite duct ve proceeds, which receives the duct of the compact white albuminiparous gland, Ed, and then becomes much enlarged, the additional width being due to the development of glandular folds, which are regarded as forming a uterus u.
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  • In Late Latin there was a tendency to this spirant pronunciation which appears as early as the beginning of the 2nd century A.D.; by the 3rd century b and consonantal u are inextricably confused.
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  • When this consonantal u (English w as seen in words borrowed very early from Latin like wall and wine) passed into the sound of English v (labio-dental) is not certain, but Germanic words borrowed into Latin in the 5th century A.D.
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  • Archiv fiir Zoologie, ii.; Id., " The Genera of European Nemerteans critically revised," Notes from the Leyden Museum (1879); Id., " Zur Anatomie u.
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  • On the influence of her cult upon that of the Virgin Mary, see Rusch, Studien u.
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  • The oxidation, which is effected by chromic acid and sulphuric acid, is conducted in a flask provided with a funnel and escape tube, and the carbon dioxide formed is swept by a current of dry air, previously freed from carbon dioxide, through a drying tube to a set of potash bulbs and a tube containing soda-lime; if halogens are present, a small wash bottle containing potassium iodide, and a U tube containing glass wool moistened with silver nitrate on one side and strong sulphuric acid on the other, must be inserted between the flask and the drying tube.
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  • This is occasioned by the y-sound with which u now begins, and is carried further in dialect than in the literary language, sue and suit, for example, being pronounced in Scotland like the Eng.
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  • The introduction of additional diacritical marks, such as - and used to express quantity, and the diaeresis, as in ai, to express consecutive vowels, which are to be pronounced separately, may prove of service, as also such letters as a, o and ii, to be pronounced as in German, and in lieu of the French ai, eu or u.
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  • U, ulna; R, radius; c, cuneiform; 1, lunar; s, scaphoid; u, unciform; m, magnum; td, trapezoid; tm, trapezium.
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  • Astrales im Weltbilde des Thalmud u.
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  • In J u d y 1909, however, the Greek flag was hoisted in Canea and Candia, and it was only lowered again after the war-ships of the protecting powefs had been reinforced and had landed an international force.
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  • Husn u 'A s4 (Beauty and Love), as his great poem is called, is an allegorical romance full of tenderness and imaginative power.
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  • The excavations at the Hieron have been recorded as they went on in the Ilpaeroat of the Greek Archaeological Society, especially for 1881-1884 and 1889, and also in the 'E4»u€pis 'ApxatoXoynoi, especially for 1883 and 1885; see also Kavvadias, Les Fouilles d'Epidaure and Tb r03'A?KX iv 'E7rukbpq, eat 9Epa7reta7'CJY Defrasse and Lechat, Epidaure.
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  • He wrote Die siebzig Jahre des Jeremias u.
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  • In 734 their king Sanip(b)u was a vassal of Tiglathpileser IV., and his successor, P(b)udu-ilu, held the same position under Sennacherib and Esarhaddon.
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  • Tubes are generally made up around mandrels, and allowed throughout the curing to remain imbedded i n p u lverized French chalk, which affords a useful support for many articles that tend to lose their shape during the process.
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  • If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.
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  • A 1, A2 � Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (al�l +a2�2) = aAa�, A 2 = (al�l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a � = a1/�1 + a21=2 respectively.
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  • If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...
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  • There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.
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  • Every covariant is rationally expressible by means of the forms f, u 2, u3,...
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  • He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.
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  • Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.
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  • The borate, Pb 2 B 6 0 1 u 4H20, is obtained as a white precipitate by adding borax to a lead salt; this on heating with strong ammonia gives PbB2044H2.
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  • 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.
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  • The grammatical forms are expressed, as in Turkish, by means of affixes modulated according to the high or low vowel power of the root or chief syllables of the word to which they are appended-the former being represented by e, o, S, ii, i l l, the latter by a, d, o, 6, u, it; the sounds e, i, i are regarded as neutral.
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  • If, for instance, we are told that 15= 4 of (x- 2), what is meant is that (I) there is a number u such that x=u+2, (2) there is a number v such that u=4 times v, and (3) 15=3 times v.
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  • From these statements, working backwards, we find successively that v= 5, u = 20, X = 22.
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  • Hence u is the greatest common measure of p and q.
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  • Assume this true for u 1, u 2, ., u,,.
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  • Then u,,+1=un+(2n-+-I)=n2+(2n+I)=(n+I)2, so that it is true for u n+1.
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  • But it is true for u 1 .
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  • 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.
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  • It vanishes when u =mlr, m being any whole number other than zero.
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  • When u = o, it takes the value unity.
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  • The maxima occur when u=tan u, (4), and then sin 2 u/u 2 = cos 2 u (5).
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  • 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.
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  • Substituting this, we find cot y = U-y, whence 5 7 y U(1+/-1-2:-+2...) - y3 -15-315' This equation is to be solved by successive approximation.
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  • It will readily be found that u =U -y =U-U-1-U-a-15U 5 105U- -.
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  • In the first quadrant there is no root after zero, since tan u> u, and in the second quadrant there is none because the signs of u and tan u are opposite.
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  • Since 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.
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  • The part corresponding to negative values of u is similar, OA being a line of symmetry.
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  • The obliquity, corresponding to u =7, is such that the phases of the secondary waves range over a complete period, i.e.
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  • Arago, the index, u for air at t° C. and at atmospheric pressure is given by 00029 iu 1-1 + 0037t.
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  • 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.
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  • 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 .
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  • Taking 2 rv2 = u (9), we may write (17r oueiudu ?/ 2) u Again, by a known formula, 1 1 °° -1 uu = 1/ 7r?r o Substituting this in (to), and inverting the order of integration, we get uc 2?
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  • Comparing 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.
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  • For example, when u = o, M = o, N =o, and consequently G =H = 2.
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  • 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.
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  • 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.
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  • Its specific gravity has the high value 18.7; its specific heat is 0.02765, which, according to Dulong and Petit's law, corresponds to U = 240: It melts at bright redness.
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  • The powdery metal when heated in air to 150° or 170° C. catches fire and burns brilliantly into U 3 0 8; it decomposes water slowly at ordinary temperatures, but rapidly when boiling.
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  • 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.
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  • Dilute sulphuric acid precipitates uranium yellow, Na 2 U 2 0 7.6H 2 O, from the solution so obtained.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • If 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.
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  • 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?
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  • 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.
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  • 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.
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  • 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.
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  • So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22, Du + dQ =o, Dv+dQ =o, Dw + dQ dt dx dt dy dt dz and taking dx, dy, dz in the direction of u, v, w, and dx: dy: dz=u: v: w, (udx + vdy + wdz) = Du dx +u 1+..
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  • 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.
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  • 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.
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  • 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.
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  • From A to B, a>u >b, 0=0, ch S2= ch log Q=cos a-i sin 2a a-b I sh S2= sh log Q= I (a u-b-a/) s i n a Q = (u-b) cos a-2(a-a') sin 2 a+1,/ (a-u.u- a')sin a (8) u-b ds _ ds d4 _ Q dw Q du - Q d 4) du q du (u-b) cos a-2(a- a') sin 2 a (a-u.0 - a') sin a (9) it j- -j' AB _f a(2b - a - a')(u-b)-2(a-b)(b-a')+2V (a - b.
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  • 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.
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  • 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'.
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  • 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 =..
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  • The integral of (14) and (15) may be written ciU+E=Fcoso, c 2 V= - Fsino, dx F cost o F sinz o 71 = U cos o - V sin o = cl + c c ic os o, chi = U sine +V coso= (F - F) sin cos o - l sino, (19) c i 2 2 2 sin o cos o - l ?
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  • 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.
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  • Thus, to take a classic example, the name of the famous king Nebuchadrezzar occurs written in the following different manners: - (a) Na-bi-urn-ku-du-ur-ri-u-su-ur,(b)AK-DU u-su-ur, (c) AK-ku-dur-ri-Shes, and (d) PA-GAR-DU-Shes, from which we are permitted to conclude that PA or AK (with the determinative for deity AN) = Na-bi-um or Nebo, that GAR-DU or DU alone = kudurri, and that Shes = u.
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  • 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.
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  • 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.
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  • In the same year appeared the first volume of the Hebrelisches u.
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  • Other works: De Pentateuchi Samaritani origine, indole, et auctoritate (1815), supplemented in 1822 and 1824 by the treatise De Samaritanorum theologia, and by an edition of Carmina Samaritana; Palaographische Studien fiber phdnizische u.
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  • 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 -.
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  • The violent tone of some of his printed manifestoes about this time, especially of his Lob des Konigs u.
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  • Of his works the more important are: - Die Composition der Genesis kritisch untersucht (1823), an acute and able attempt to account for the use of the two names of God without recourse to the document-hypothesis; he was not himself, however, permanently convinced by it; De metris carminum Arabicorum (1825); Das Hohelied Salomo's Ubersetzt u.
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  • Jakobos' Rundschreiben (1870); Die Lehre der Bibel von Gott, oder Theologie des alien u.
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  • 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.
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  • This arises from the pronunciation of u as yu, and does not affect the English dialects which have not thus modified the u sound.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • It is easy to arrange the experiment so that U is small and nearly constant.
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  • 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.
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  • Scholz, Hubert Languet als kursachsischer Berichterstatter u.
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  • 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.
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  • The os magnum 1, lunar; sc, scaphoid; u, unciform; of the carpus articulates freely m, magnum; td, trapezoid; tm, with the scaphoid.
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  • 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.
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  • 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'.
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  • In one, applicable only to liquids which do not mix, the two liquids are poured into the limbs of a U tube.
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  • The whole micrometer-box is moved by u? ??'
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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 +...
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  • Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respectively.
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  • To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x.
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  • The ordinate of the trapezette will be denoted by u, and the abscissa of this ordinate, i.e.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • The simplest case is that in which u is constant or is a linear function of x, i.e.
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  • The next case is that in which u is a quadratic function of x, i.e.
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  • 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).
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  • 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).
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  • 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.
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  • When u is of degree 4 or 5 in x, we require at least five ordinates.
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  • 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).
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  • 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.
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  • 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.
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  • 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+..
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  • 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.
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  • 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.
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  • 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.
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  • The process is simplified by writing down the general formula first and then substituting the values of u.
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  • 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.
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  • 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.
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  • If the data are uo, u 1,.
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  • If the data are u;, U I, ...
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  • The justification of the above methods lies in certain properties of the series of successive differences of u.
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  • 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).
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  • If we do not know values of u outside the figure, we must use advancing or receding differences.
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  • To find the pth moment, when uo, u l, u 2, ...
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  • 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"',.
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  • 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.
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  • 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),.
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  • 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, ..
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  • Parinentier's rule, for instance, assumes that in addition to u I; u I..
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  • 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'?
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  • 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.
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  • The Euler-Maclaurin formula (§ 75) assumes that the bounding values of u', u"',..
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  • 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.
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  • 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.
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  • 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.
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  • Attention must be given to the possible accumulation of errors due to the small errors in the values of u.
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  • 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.
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  • 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.
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  • 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.
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  • But it has velocity U, and therefore momentum poU 2 is carried in.
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  • 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.
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  • 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.
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  • Equating this to the momentum entering at B and subtracting P' from each X+W+p(U - u)2 =poU 2.
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  • 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.
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  • 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.
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  • 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).
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  • 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.
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  • 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).
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  • 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].
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  • At 0° C.: we have Uo =1/ (yk), and hence U t = Uolt (I +at) =U 0 (I+-o o0184t) (for small values of t).
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  • In the momentum equation (4) we may now omit X and it becomes 0.+P(U - u) 2 =poU2.
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  • We have U - u =U(I - u/U) =U(1 - v/V), since u/U= - dy/dx= v/V.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • We also have pu t =p o u t /(I +dy/dx).
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  • We have U-Fw= D/T 1 and U - w = D/T .
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  • The theoretical investigation given above shows that if U is the velocity in air at 1° C. then the velocity U ° at o° C. in the same air is independent of the barometric pressure and that Uo = U /(1 +o o01841), whence U 0 =332 met./sec.
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  • 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 }.
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  • The velocity deduced at 8.1° C. was U=1435 met./sec., agreeing very closely with the value calculated from the formula U 2 = E/p.
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  • 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.
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  • Let S' be its position one second later, its velocity being u.
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  • Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.
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