## Dt Sentence Examples

- During his struggles with the Saxons; he fought for Henry at Warnsta,
**dt**and was killed in his service at Welfesholz. - If I be the change in the internal energy, the relation referred to gives us the equation A = I +T (dA/
**dT**), where dA/**dT**denotes the rate of change of the available energy of the system per degree change in temperature. - Hence we get the equation Ee = Le +Te (dE/
**dT**) or E = L+T(dE/**dT**), as a particular case of the general thermodynamic equation of available energy. - It will be noticed that when dE/
**dT**is zero, that is, when the electromotive force of the cell does not change with temperature. - According to this notation, the three equations of motion are dt2 = b2v2E + (a2 - b2) d.s
**dt**=b2v2rj+(a2 - b2) dy d2 CIF - b2p2+(a2_b2)dz It is to be observed that denotes the dilatation of volume of the element situated at (x, y, z). - Phil.): - Let x, y, z be the coordinates of P in the orbit,, r t, those of the corresponding point T in the hodograph, then dx dy _ dz c= ' 71 - a' - at therefore Also, if s be the arc of the hodograph, ds = v = V V1 1) j
**dt**+ (dt2)**dt**Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration. - Giessen, Altdorf, Helmsta.
**dt**, Jena, Wittenberg), as well as to Upsala in Sweden. - Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and 0 the angle which the outward-drawn normal makes with the velocity q at that point, dM/
**dt**= rate of increase of fluid inside the surface, (I) =flux across the surface into the interior _ - f f pq cos OdS, the integral equation of continuity. - The integral equation of continuity (I) may now be written l f fdxdydz+ff (lpu+mpv+npdso, (4) which becomes by Green's transformation (
**dt**+d dz dy dx (p u) + d (p v) + d (p w) l I dxdydz - o, dp leading to the differential equation of continuity when the integration is removed. - The time rate of increase of momentum of the fluid inside S is )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that /if (dpu+dpu2+dpuv +dpuw_ +d p j d xdyd z = o, (b)`
**dt**dx dy dz dx / leading to the differential equation of motion dpu dpu 2 dpuv dpuv _ X_ (7)**dt**+ dx + dy + dz with two similar equations. - These equations may be simplified slightly, using the equation of continuity (5) § for dpu dpu 2 dpuv dpuw
**dt**dx + dy + dz =p Cat +uax+vay+waz? - Dp dpu dpv dpw -z)' reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual form du du du du d
**dt**+udx+vdy +wdz =X -n dx' with the two others dv dv dv dv i dp**dt**+u dx +v dy +w dz - Y -P d y' dw dw dw Z w dw i d p**dt**+u dx +v dy +wd - -P dz. - To determine the component acceleration of a particle, suppose F to denote any function of x, y, z, t, and investigate the time rate of F for a moving particle; denoting the change by DF/
**dt**, DF = 1t F(x+uSt, y+vIt, z+wSt, t+St) - F(x, y, z, t)**dt**at = d + u dx +v dy+ w dz and D/**dt**is called particle differentiation, because it follows the rate of change of a particle as it leaves the point x, y, z; but dF/**dt**, dF/dx, dF/dy, dF/dz (2) represent the rate of change of F at the time t, at the point, x, y, z, fixed in space. - (5) (8) (I) The components of acceleration of a particle of fluid are consequently Du dudu du du
**dt**=**dt**+u dx +v dy + wdz' Dr dv dv dv dv**dt**-**dt**+udx+vdy+wdz'**dt**v =**dtJ**+udx+vdy +w dx' leading to the equations of motion above. - 2wr { a 0, dt2WE+2UC+ dz = o, dw
**dt**- 2un+2v+ dH = 0, where H = fdp/p +V +1q 2, (7) 2 2 +v 2 2 (8) and the three terms in H may be called the pressure head, potential head, and head of velocity, when the gravitation unit is employed and Zq 2 is replaced by 1q 2 1 g. - Eliminating H between (5) and (6) DS du dv dw (du dv d1zv
**dt**u dx n dx udx' 5 -, dzi =°' and combining this with the equation of continuity Dp du dv dw p iit dx+dy+ dz = °' (10) D i du n dv dw_**dt**(p p dx p dx p dx - o, with two similar equations. - Putting do dipdp _ Hdt -
**dt**-K, the equations of motion (4), (5), (6) § 24 can be written -2u +2wn - d(x',t))o,...,. - Equation (5) becomes, by a rearrangement, dK dmdm dm din dx
**dt**+u dx + dy +Zee dz + dx (**dt**+u dx +v dy +w d) = o,. - Dm dm Dl (8), dx - dx
**dt**+ dx**dt**= °' ...' - With liquid of density p, this gives rise to a kinetic reaction to acceleration dU/
**dt**, given by 7rp b 2 a 2 b l b d J = a 2 +b2 M' dU, if M' denotes the mass of liquid displaced by unit length of the cylinder r =b. - 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+.. - Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are 1, m, n, k=lu+mv+nw; (2) and in the infinitesimal element of time
**dt**, the coordinates of the fluid particle at (x, y, z) will have changed by (u', v', w')**dt**; so that Dk dl, do**dt****dt****dt****dt**+**dtw**+1 (?t +u, dx +v, dy +w, dz) +m (d +u dx + v dy +w' dz) dw, dw +n (**dt**?dx+v?dy +w dz) But as 1, m, n are the direction cosines of a line fixed in space, dl= m R-n Q, d m = nP-lR an =1Q-mP**dt****dt**'**dt**' so that Dk __ du, du, du, du**dt**l (**dt**-vR+ wQ+u + v dy + w dz) +m(.. - U '= - dx -md x, ' - dy -m dy, w = - dz-mdz' as in § 25 (I), a first integral of the equations in (5) may be written dp V + 2q 2 - d - n
**dt**+14-14) (dx + m dz) +(v-v') (+m) +(w - w) (+m) =F(t), (7) in which d4, do, d? **Dt**-(u)dy- (w-w) dz = d - (U-yR+zQ) dy - (V-zP+xR)d -(W-xQ+yP) d z (8) is the time-rate of change of 49 at a point fixed in space, which is left behind with velocity components u-u', v-v', w-w'.- In the general motion again of the liquid filling a case, when a = b, 52 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d =- 2+ 2 J, = 2 x22111, d = 2 2`2 (+/'15-Om) (1 yy y n`t
**dt**a +c**dt**a +c**dt**a +c) dc2, a2-1-c2 d122 a2 c2**dt**="2) +a2= G2y 71'**dt**= 121 1 - a 2 -c 2SJ, (19) of which three integrals are e +777 r z y 2= L -?2J2, (20) (a2 + c2) 2 2 121+14 =M+ 2c2(a2-c2)1 ' (21) 121+522hN = + x24 2,2 and then (**dt**/ 2 = (a2 + c 2) 2(° v 2 - 12171) 2 4C4 2 2 - (+ c2)2?(E+77) (? - Put S2 1 =12 cos 4, 12 2 = -12 sin 4, d4 d52 1 dS22 Y a2+c2 122 7Ti = 71 22 CL2- c2(121+5221)J, a2 +c2 do a2+c2 + 4c2 z
**dt**a'-c2 (a2+,c2)2 M+2c2(a2-c2 N-{-a2+c2 2 Ý_a 2 +c 2 (' 4c2 .?"d za 2 -c 2 c2)2 2'J Z M+ -c2) which, as Z is a quadratic function of i 2, are non-elliptic so also for; G, where =co cos, G, 7 7 = - sin 4. - In a state of steady motion d4- 121 _S22 Tit °' - fl 4=1G = nt, suppose, S21 -F9,277 = S2co, d4 a2+c2 WI- 1 a2-c2S21' _ 2a 2 SZ
**dt**a2+c2cos' a 2 + c 2 a, 2 a 2 S2 I- a2_c22--a2+C2,0, 1a2 c2)2 (a 2 -c 2) (9a2-c2) ? - M _ c Q du dO
**dt**- rrqu 2r qu AB _ Q du L bq (u -a') +V (b -a')1,l (a -u)11/ndu) L 1,1 (a-a/),,1 (u-b), f u Along the wall Bx, cos n0 =I, sin n0 =o, b >u>o ch nSt = ch log () n=, , fb-a?, ? - 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 =.. - 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 =.. - These equations are proved by taking a line fixed in space, whose direction cosines are 1, then
**dt**=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP**dt**xl+, d y t x2' x3 +1**dtl****dt**2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed. - Let E be the effective elasticity of the aether; then E = pc t, where p is its density, and c the velocity of light which is 3 X 10 10 cm./sec. If = A cos" (t - x/c) is the linear vibration, the stress is E dE/dx; and the total energy, which is twice the kinetic energy Zp(d/
**dt**) 2 dx, is 2pn2A2 per cm., which is thus equal to 1.8 ergs as above. - A dielectric substance is electrically polarized by a field of electric force, the atomic poles being made up of the displaced positive and negative intrinsic charges in the atom: the polarization per unit volume (f',g',h') may be defined on the analogy of magnetism, and d/
**dt**(f',g',h') thus constitutes true electric current of polarization, i.e. - These circuital relations, when expressed analytically, are then for a dielectric medium of types = (
**dt**+ x) (f',g',h')+**dt**(f,g,h), dR dQ = da dy dz**dt**' ' I See H. - For the simplest case of polarized waves travelling parallel to the axis of x, with the magnetic oscillation y along z and the electric oscillation Q along y, all the quantities are functions of x and t alone; the total current is along y and given with respect to our moving axes by __ (d_ d Q+vy d K-1 Q,
**dt**dx) 47rc 2 +**dt**(4?rc 2) ' also the circuital relations here reduce to _ dydQ _dy _ dx 47rv ' _**dt**' d 2 Q dv dx 2 -417t giving, on substitution for v, d 2 Q d 2 Q d2Q (c2-v2)(7372 = K**dt**2 2u dxdt ' For a simple wave-train, Q varies as sin m(x-Vt), leading on substitution to the velocity of propagation V relative to the moving material, by means of the equation KV 2 + 2 uV = c 2 v2; this gives, to the first order of v/c, V = c/K i - v/K, which is in accordance with Fresnel's law. - Since no angular momentum goes out on the whole Z nwra 4 d0/dx -?- 2 pwra 4 Ud0/
**dt**= o. - Rods of different materials may be used as sounders in a Kundt's dust tube, and their Young's moduli may be compared, since: length of rod Then dO U = - ax dx or
**dt**= - UK. - In the time
**dt**which the wave takes to travel over MN the particle displacement at N changes by QR, and QR= - udt, so that QR/MN = - u/U. - 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**. - On in time
**dt**to A'B'C', where AA' = Udt, the displacement of the particle originally at M must change from PM to P'M or by PP'. - If then dl is an element of the path, putting
**dt**= do/U, we have the average excess of pressure p = ° (to } pu 2)**dt**= Lj j (+pu2)dI. - The velocity perpendicular to the axis of any point on the curve at a fixed distance x from 0 is dy_ (I ]) at A A The acceleration perpendicular to the axis is d2y = 2 2
**dt**2 - A 2 sin A (x - Ut) The maximum pressure excess is the amplitude of ("6= Eu /U _ (E/U)dy/**dt**.