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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).

00dW 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.

00dp 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.

002wr { 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.

00Eliminating 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.

00So 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+..

00Uniplanar 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.

00- 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.

007), 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.

00u -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.

00From 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.

003), 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'.

00The 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.

00He 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.

00If 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.

00Since Od4 cannot be less than dH, the difference (61d4-dE) cannot be less than dW.

00The condition in this form can be readily applied provided that the external work dW can be measured.

001, w; 2, .fn; 3, /ImI; 4, fdw; 5, dw; 6, sls (or Sw.?); 7, sfii; 8, llmn; 9, ps~ 10, ml.

00If 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.

00The lowest quality of Riga flax is marked DW, meaning Dreiband Wrack.

00Integrating 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.

00DW: Do you ride a bike much yourself, ever done a timed run down the Nevis Range track?

00DW: Proper kilt for Fort Bill in 05 rather than this year's tartan car blanket?

00DW: Who will take over your mantle as junior champ?

00DW: Is the vibe chilled out at the Masters, or more serious than a Norba?

00DW: Finally Vanessa, how much wood would a woodchuck chuck chuck if a woodchuck could chuck wood?

00BSRIA Testing and Certification is able to evaluate the performance of rectangular ductwork flanged duct joints to HVCA standard DW / TM1: 1987.

00DW: Current choice between motorcycle helmet or bike helmet?

00Plain cream printed dw, black lettering on front & spine, not price clipped 3s 6d net printed on front.

00DW: I hear you make a delicious banana loaf can we try some?

00I use the DW 9000 double bass drum pedals.

00In late February 2004, the large planetoid labeled 2004 DW was found in Edgeworth-Kuiper Belt (10 ).

00DW: Finally Vanessa, how much wood would a woodchuck chuck if a woodchuck chuck if a woodchuck could chuck wood?

00The 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 Ã‚° (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).

00dW 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.

00dp 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.

00To integrate the equations of motion, suppose the impressed force is due to a potential V, such that the force in any direction is the rate of diminution of V, or its downward gradient; and then X= -dV/dx, Y= -dV/dy, Z= -dV/dz; (I) and putting dw dv du dw dv du Ty - dz -2 ' dz - dx -2n ' dx - dy2?, d -{- d ' v ?

002wr { 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.

00Eliminating H between (5) and (6) DS du dv dw (du dv d1zv dt u dx n dx udx' 5 -, dzi =Ã‚°' and combining this with the equation of continuity Dp du dv dw p iit dx+dy+ dz = Ã‚°' (10) D i du n dv dw_ dt (p p dx p dx p dx - o, with two similar equations.

00So 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+..

00Now 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(..

00Uniplanar 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.

00- 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.

00(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') _ ?

007), 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.

00u -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.

00From 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.

003), 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'.

00The 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.

00He 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.

00If 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.

00Since Od4 cannot be less than dH, the difference (61d4-dE) cannot be less than dW.

00The condition in this form can be readily applied provided that the external work dW can be measured.

001, w; 2, .fn; 3, /ImI; 4, fdw; 5, dw; 6, sls (or Sw.?); 7, sfii; 8, llmn; 9, ps~ 10, ml.

00If 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.

00The lowest quality of Riga flax is marked DW, meaning Dreiband Wrack.

00Then if 0 is the centre of curvature in the plane of the paper, and BO =u, I _ cos sinew u R 1 R2 Let POQ=o, PO=r, PQ=f, BP=z, f 2 = u 2 +r 2 -2ur cos 0 (26) The element of the stratum at Q may be expressed by ou t sin o do dw, or expressing do in terms of df by (26), our 1fdfdw.

00Integrating 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.

00an d „...d---: e 'a ' ~ t a aas?Caed; C dw ick en h ornlury LI?'.

00DW: All homeowners are required to have homeowners insurance if they have a mortgage on their house.

00DW: Mortgage insurance simply protects the bank's financial interest for the amount of the outstanding loan.

00DW: It depends what that $250,000 includes.

00DW: Choose a deductible at the limit that you no longer can comfortably afford to pay out of your own pocket.

00DW: Outbuildings (or any unattached building) normally are covered at 10 percent of the dwelling value.

00DW: Your contents are covered for about 70 percent of your dwelling value (depending on the insurance company).

00DW: As a consultant, I don't sell insurance.

00DW: Always take the time to shop around on your own among at least three agents.

00

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