## Dp Sentence Examples

**Dp**, Dactylopore.- If we denote the critical volume, pressure and temperature by Vk, Pk and Tk, then it may be shown, either by considering the characteristic equation as a perfect cube in v or by using the relations that
**dp**/dv=o, d 2 p/dv 2 =o at the critical point, that Vk = 3b, Pk= a/27b2, T ic = 8a/27b. - +mp4dp4 +...) =exp Mp g
**dp**4+ï¿½ï¿½ . - Recalling the formulae above which connect s P4 and a m, we see that dP4 and
**Dp**q are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10 P 01 4) respectively. - But by Green's transformation f flpdS = f f PPdxdydz, (2) thus leading to the differential relation at every point = dy
**dp**The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically. - Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then
**dP**dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential. - As an example of the general equations, take the simplest case of a uniform field of gravity, with Oz directed vertically downward; employing the gravitation unit of force, 1
**dp**i**dp**t**dp****dp**/dz = p = pzn (4) z n+I pz 1 /n p-p n-H ?t), (5) supposing p and p to vanish together. - From the gas-equation in general, in the atmosphere n d
**dp**_ I**dp**1 de _ d0 de i de (8) z p dz-edz-p-edz-k-edz' which is positive, and the density p diminishes with the ascent, provided the temperature-gradient de/dz does not exceed elk. - With uniform temperature, taking h constant in the gas-equation,
**dp**/ dz= =p / k, p=poet/ k, (9) so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p 1 and 1, 2 at heights z 1 and z2 (z1-z2)11? - Then
**dp**/dz=kdp/dz = P, = Poe ik, p - po= kpo(ez Ik -1); (16) and if the liquid was incompressible, the depth at pressure p would be (p - po) 1po, so that the lowering of the surface due to compression is ke h I k -k -z= 1z 2 /k, when k is large. - On another physical assumption of constant cubical elasticity A,
**dp**= Ad p /P, (p - po)IA= lo g (P/Po), (18)**dp**_ A**dp**(I 1 zd p dz - P ' A Po-p -z, I - p -k, A kPo ' (19) (3) P dx Pdy Pdz -., (I) When the density p is f un**dp**/ iform, this becomes, as before in (2) § 9 P pp ==Pzz++pao constant. - 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. **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.- 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. - If r denotes the radius of curvature of the stream line, so that I
**dp**+ dV - dH _ dq 2 q2 (6) p dv dv dv dv - r ' the normal acceleration. - 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? - The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, become
**dp**+4 p Ax+ du - vR {-wQ? - 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. - In the notation of the calculus the relations become - dH/
**dp**(0 const) = odv /do (p const) (4) dH/dv (0 const) =odp/do (v const) The negative sign is prefixed to dH/**dp**because absorption of heat +dH corresponds to diminution of pressure -**dp**. The utility of these relations results from the circumstance that the pressure and expansion co efficients are familiar and easily measured, whereas the latent heat of expansion is difficult to determine. - Dividing by (0' - e"), and writing
**dp**/do and dL/do for the limiting values of !the ratios (p' - p")/(o' - o") and (L' - L")/(o' - o"), we obtain the important relations s' - s"+dL/do= (v" - v')**dp**/do=L/o,.. - (5) in which
**dp**/do is the rate of change of pressure with temperature when the two states are in equilibrium. - Substituting for H its value from (3), and employing the notation of the calculus, we obtain the relation S - s =0 (
**dp**/do) (dv/do),. - It is often impossible to observe the pressure-coefficient
**dp**/de directly, but it may be deduced from the isothermal compressibility by means of the geometrically obvious relation, BE = (BEÃ†C) XEC. The ratio BEÃ†C of the diminution of pressure to the increase of volume at constant temperature, or -**dp**/dv, is readily observed. - We thus obtain the expressions dH = sdo +0 (
**dp**I dO) dv = Sd0 - o (dv/do)**dp**.. - If we put dH=o in equations (8), we obtain the relations between dv and do, or
**dp**and do, under the condition of no heat-supply, i.e. - (9) These must be expressed as functions of v and 0, which is theoretically possible if the values of s, p, and
**dp**/do are known. - Observing that F is a function of the co-ordinates expressing the state of the substance, we obtain for the variation of S with pressure at constant temperature, dS/
**dp**(0 const) '=' 2 F/dedp =-0d 2 v/d0 2 (p const) (12) If the heat supplied to a substance which is expanding reversibly and doing external work, pdv, is equal to the external work done, the intrinsic energy, E, remains constant. - The isothermal elasticity - v(
**dp**/dv) is equal to the pressure p. The adiabatic elasticity is equal to y p, where -y is the ratio S/s of the specific heats. - The heat absorbed in isothermal expansion from vo to v at a temperature 0 is equal to the work done by equation (8) (since d0 =o, and 0(
**dp**/d0)dv =pdv), and both are given by the expression RO log e (v/vo). - In thiscase the ratio of the specific heats is constant as well as the difference, and the adiabatic equation takes the simple form, pv v = constant, which is at once obtained by integrating the equation for the adiabatic elasticity, - v(
**dp**/dv) =yp. - It is found by experiment that the change of pv with pressure at moderate pressures is nearly proportional to the change of p, in other words that the coefficient d(pv)/
**dp**is to a first approximation a function of the temperature only. - (15) where d0 is the fall of temperature of the fluid corresponding to a diminution of pressure
**dp**. If there is no fall of temperature in passing the plug, d0 = o, and we have the condition Odv/d0 =v. - Trans., 1854, 1862) found that the cooling effect, do, was of the same order of magnitude as the deviations from Boyle's law in each case, and that it was proportional to the difference of pressure,
**dp**, so that d0/**dp**was nearly constant for each gas over a range of pressure of five or six atmospheres. - Experiments by Natanson on CO 2 at 17° C. confirm those of Joule and Thomson, but show a slight increase of the ratio do/
**dp**at higher pressures, which is otherwise rendered probable by the form of the isothermals as determined by Andrews and Amagat. - The value of the angular coefficient d(pv)/
**dp**is evidently (b - c), which expresses the defect of the actual volume v from the ideal volume Re/p. Differentiating equation (17) at constant pressure to find dv/do, and observing that dcldO= - nc/O, we find by substitution in (is) the following simple expression for the cooling effect do/**dp**in terms of c and b, Sdo/**dp**= (n+I)c - b.. - (25) „ (11) dF/
**dp**„ = n+i)c - b . - (26) „ (10) ds/dv (I - n+2nc/V)Rnc/V2 (27) „ (12) dS/
**dp**„ =n(n+I)c/e. - In the case of an ideal gas,
**dp**/d9 at constant volume =R/v, and dvld6 at constant pressure =R/p; thus we obtain the expressions for the change of entropy 0-4)0 from the state poeovo to the state pev, log e e/eo+R logev/vo =S log e 9/00-R (32) In the case of an imperfect gas or vapour, the above expressions are frequently employed, but a more accurate result may be obtained by employing equation (17) with the value of the specific heat, S, from (29), which gives the expression 4-¢o = Sologe0/00 - R logep/po-n(cp/B-copo/Bo) - Assuming the function G to be expressed in terms of p and 0, this condition represents the relation between p and 0 corresponding to equilibrium between the two states, which is the solution of the relation (v" - v')
**dp**/dO=L/D, (5). - (39) dG"/
**dp**(D const) =v, dJ"/dv (0 const) = p. (40) And all the properties of the substance may be expressed in terms of G or J and their partial differential coefficients. - Let the product
**dp**i dq i ... **Dp**n dq n be spoken of as the " extension " of this range of values.- 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). - From equations (i), we find that aq _ o a pi qi -, so that the extension of the new range is seen to be
**dp**i dq i ... - Since by definition E_ - v(
**dp**/dv) =p(**dp**/**dp**) equation (6) becomes U = (**dp**/**dp**) (7) The value U = / (E/p) was first virtually obtained by Newton (Principia, bk. - 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). - The Bessel function of order m satisfies the differ ential equation
**dP**2 -}- p - -}- (I _!!) u =o, and may be expressed as the series 2 l**dp**2 2 + 2 -? - Hence
**dPo**=**dP**' -**dP**and**dPo**/**dP**=(V - V')/V' or**dPo**/**dP**' = (V - V')/V. - V'
**dP**' = and (P VdP = C Po vdp, whence J f P' V'**dP**' - (P Vd .1 P = (P v?**dp**, I?'