## DW Sentence Examples

- The small magnet may be a sphere rigidly magnetized in the direction of Ho; if this is replaced by an isotropic sphere inductively magnetized by the field, then, for a displacement so small that the magnetization of the sphere may be regarded as unchanged, we shall have
**dW**= - vIdHo = v I+-, whence W = - 2 I + H2 ° (37) The mechanical force acting on the sphere in the direction of displacement x is 1 Hopkinson specified the retentiveness by the numerical value of the " residual induction " (=47rI). **DW**F - d -v 1+ a 7rK dx dH (38) (34) [[[Magnetic Measurements]] If Ho is constant, the force will be zero; if Ho is variable, the sphere will tend to move in the direction in which Ho varies most rapidly.- 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. - 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. - 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(.. - Uniplanar motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the coordinates x, y by w=f(z), where z= x +yi, w=4-Plg, and then
**dw**dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q**dw**u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a vector representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q. - - n)= l b - au - ' (8) a - a u - b (9) dS2 I A I (b-a.b-a')
**dw**m du = 21/(U - b)- ‘ 1 (u-a.0-a')' du -, r u' Io) the formulas by which the conformal representation is obtained. - (12) Along the stream line xBAPJ, t ' =0, u=ae-" c bl, n; (13) and over the jet surface JPA, where the skin velocity is Q, - q = - Q, u = ae rs Q /m = ae rs lc, (14) ds denoting the arc AP by s, starting at u = a; a ' ch nS2=cos nB= -a' u u - - a b' (15) a l a - b l u - a' a-a' u-b' co > u = ae'" S " c > a, and this gives the intrinsic equation of the jet, and of curvature ds '&1) _ i
**dw**i**dw**dS2 P= - dO = Q a0 - Q as2 = Q c u-b d (u -a.u -a') _ ? - 7), and so must be excluded from the boundary of u; the conformal re presentation is made now with du= (b-a.b-a') du - (u-b) A l (u-a.0-a) (I)
**dw**m I m' du = 7r u-j - u -j' _ m+m' u-b it u' j.0-j" b = mj i m'j m+m', taking u = co at the source where FIG.7. - U -b' Along a jet surface, q=Q, and ch S2= cos 0 =cos a-i sin2a(a-a')/(u-b), (5) if 0 =-a at the source x of the jet xB, where u = co; and supposing 0=0,13 at the end of the streams where u =j, j', u-b i sin 2 a u - j cos 0-cos /3 i a -a cos a sin a -cos 0' aa' - 2 (cos a -cos (3) (cos a-cos 0)' u-j' 1 2 cos 0-cos, (6) a -a' - 2 S i n a (cos a -cos (3') (cos a -cos B)' and 4' being constant along a stream line d4 -
**dw**ds _d8 d4 _**dw**du du du' d- -dud0' 7rQ ds_ it ds (cos a-cos /3) (cos a -cos (3') sin 0 m+m' dB c d0 - (cos a-cos B) (cos 0-cos /3) (cos 0 -cos /3')' _ sin 0 cos a-cos 13 sin 0 - cos a-cos B + cos 0-cos (3' cos 0-cos 13 cos a -cos $ sin 6 cos (3-cos /3' cos 0-cos 0" giving the intrinsic equation of the surface of a jet, with proper attention to the sign. - 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. - 3), and describe through it as centre a cone of small solid angle
**dw**cutting out of the enclosing surface in two small areas dS and dS' at distances x and x'. - The normal section of the cone at that point is equal to dS cosO, and the solid angle
**dw**is equal to dS cos0/x 2. - 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. - With this definition of temperature 0, if the heat H is measured in work units, the expression of Carnot's principle for an infinitesimal cycle of range do reduces to the simple form
**dW**/d9=H/0. - If
**dW**is the external work done, dH the heat absorbed from external sources, and dE the increase of intrinsic energy, we have in all cases by the first law, dH-dE=**dW**. - The condition in this form can be readily applied provided that the external work
**dW**can be measured. - Integrating with respect to f from f =z to f=a, where a is a line very great compared with the extreme range of the molecular force, but very small compared with either of the radii of curvature, we obtain for the work (1,G (z) - 111(a))
**dw**, and since (a) is an insensible quantity we may omit it. - The lowest quality of Riga flax is marked
**DW**, meaning Dreiband Wrack. - If
**dw**/dt is the angular acceleration of the link,**dw**/dt X CB is the tangential acceleration of the point B about the point C. Generally this tangential acceleration is unknown in magnitude, and it becomes part of the problem to find it. - If X, Y, Z are the components of force, then considering the changes in an infinitely short time 3t we have, by projection on the co-ordinate axes, i3(mu) =Xi5t, and so on, or du dv
**dw**m-~jj=X, m~=Y, m~=Z. - 1, w; 2, .fn; 3, /ImI; 4, fdw; 5,
**dw**; 6, sls (or Sw.?); 7, sfii; 8, llmn; 9, ps~ 10, ml.