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).
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.
C or cz is pronounced as English ts; cs as English ch; ds as English j; zs as French j; gy as dy.
Consider the integral W dx dy dz .
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.
Hence the total flux is - (+ d2V d 2 V d2V dye + dz2) dy dz, dx2 and by the previous theorem this must be equal to 4'rrp dxdydz.
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'?
If the liquid is stirred up by the rotation R of a cylindrical body, d4lds = normal velocity reversed dy = - Rx- Ry ds (5) ds 4' + 2 R (x2 + y2) = Y, (6) a constant over the boundary; and 4,' is the current-function of the relative motion past the cylinder, but now V 2 4,'+2R =o, (7) throughout the liquid.
If we suppose that the force impressed upon the element of mass D dx dy dz is DZ dx dy dz, being everywhere parallel to the axis of Z, the only change required in our equations (I), (2) is the addition of the term Z to the second member of the third equation (2).
If we write -fxo f yox s yiu dx dy, we first calculate the raw values coo., ai,o, 0.1,1,
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.
In the case of a steady motion of homogeneous liquid symmetrical about Ox, where 0 is advancing with velocity U, the equation (5) of § 34 p/p +V +Zq'2-f(,P') = constant becomes transformed into P +V + 2- dy + 2U 2 -f(t +2Uy 2) = constant, = 1,t+4Uy2, subject to the condition, from (4) § 34, Y -2 V = - f ' (Y', y 2 2 +2Uy2).
In the equations of uniplanar motion = dx - du = dx + dy = -v 2 ?, suppose, so that in steady motion dx I +v24 ' x = ?'
Let this rectangular prism be supposed to be wholly filled up with electricity of density p; then the total quantity in it is p dx dy dz.
Let us apply the above theorem to the case of a small parallelepipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z).
Since the dimensions of T are supposed to be very small in com d parison with X, the factor dy (--) is sensibly constant; so that, if Z stand for the mean value of Z over the volume T, we may write TZ y d e T ?
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.
The 18KT gold posts are stamped with the European 750 symbol (indicating 18KT gold) and the trademark initial DY.
The integration of the several terms may then be effected by the formula e y dy =r(4+2)=(4 - i)(4-2)...
The osculating plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are du du, du d i g d g 2 _ dH dx +v dy + dz - 2v?
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.
There is, however, considerable evidence in support of the view that Greek va representing the sound arising from Ky, xy, Ty, By was pronounced as sh (s), while representing gy, dy was pronounced in some districts zh (z).4 On an inscription of Halicarnassus, a town which stood in ancient Carian territory, the sound of vv in `AXoKapvaao-Ewv is represented by T, as it is also in the Carian name Panyassis (IIavvfiTcos, geni tive), though the ordinary is also found in the same inscription.
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?
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 =..
When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = o, and C reduces to C = ff cos px cos gy dx dy,.
Without attempting to answer this question categorically, it may be pointed out that within the limits of the family (Ptychoderidae) which is especially characterized by their presence there are some species in Y art dY YY cts, posterior limit of collar.