## 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 =.. - 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. - Origin up to the moving origin 0, so that dy x=y=z=o, but
**dt**U,**dt**= ' dG _ dyl =l (- yi y3Q x2w+xiv) +m (dY2yP+Yrxu+xw) +n (? - Clebsch to take the form T= 2p(x12 +x22)+2p'x32 + q (xiyi +x2y2) +q'x3y3 +2r(y12+y22)+2r'y32 so that a fourth integral is given by dy 3 /
**dt**= o, y = constant; dx3 (4 y) (q + y) _ (y y)**dt**- xl 'x2 xl Y Y x l 2 - 1, y2 () = (x12 +x22) (y12 + y22) = (X 1 2 + X 2) +y22)-(FG-x3y3)2 = (x 1 y32-G2)-(Gx3-Fy3) 2, in which 2 = F 2 -x3 2, x l y l +x2y2 = FG-x3y3, Y(y1 2 +y2 2) = T -p(x12 +x22) -p'x32 -2q(xiyi 'x2y2)- 2 q ' x = (p -p') x 2 + 2 (- q ') x 3 y 3+ m 1, (6) m1 = T 2 i y 3 2 (7) so that dt3) 2 =X3, (8) where X3 is a quartic function of x3, and thus t is given by an elliptic (8) (6) (I) integral of the first kind; and by inversion x 3 is in elliptic function of the time t. - Introducing Euler's angles 0, c15, x1= F sin 0 sin 0, x 2 =F sin 0 cos 0, xl+x 2 i =iF sin 0e_, x 3 = F cos 0; sin o t=P sin 4+Q cos 0,
**dT**F sin 2 0d l - dy l + dy 2x = (qx1+ryi)xl +(qx2+ry2)x2 = q (x1 2 +x2 2) +r (xiyi +x2y2) = qF 2 sin 2 0-Fr (FG - x 3 y 3), (16) _Ft (FG _x 323 Frdx3 (17) F x3 X3 elliptic integrals of the third kind. - Sin o= F dl, (20) C3 do F2 h _ F2 cos 2 o F 2 sin z o F
**dt**y - V C G c +2 c1 coso+H]; (21) 1 z so that cos 0 and y is an elliptic function of the time. - When is absent, dx/
**dt**is always positive, and the centre of the body cannot describe loops; but with E, the influence may be great enough to make /**dt**change sign, and so loops occur, as shown in A. - He therefore employed the corresponding expression for a cycle of infinitesimal range
**dt**at the temperature t in which the work dW obtainable from a quantity of heat H would be represented by the equation dW =HF'(t)**dt**, where F'(t) is the derived function of F(t), or dF(t)/**dt**, and represents the work obtainable per unit of heat per degree fall of temperature at a temperature t. - After a time
**dt**the value of p i will have increased to pi+pidt, where p i is given by equations (i), and there will be similar changes in qi, P2, q2, ... - Thus after a time
**dt**the values of the coordinates and momenta of the small group of systems under consideration will lie within a range such that pi is between pi +pidt and pi +dp,+(pi+ap?dpi)**dt**„ qi +gidt „ qi+dqi+ (qi +agLdgi)**dt**, Thus the extension of the range after the interval**dt**is dp i (i +aidt) dq i (I +?gidt). - Or, expanding as far as first powers of
**dt**, dpidqi. - 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. - On combining the two parts of the pressure which have been calculated, the aggregate impulsive pressure on dS in time
**dt**is found to be dS ff f v1,1 (h 3 m 3 /7r 3)e h m(u2+v2+w2 >mu2dudvdw, where M denotes summation over all kinds of molecules. - 5(plan) at a range HT, if the axis were directed on T, drift would carry the shot to D, therefore the axis must be directed on a point D' such that D'T =
**DT**. - Now if the notch of the tan gent sight be carried to H' in order to lay on T, the fore-sight, and with it the axis, H will be moved to F', the line of fire will be HF'D', and the shot will strike T since D'T =
**DT**. - 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**.