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dz

Thus so that z 1 J1 2 (z) = - 2 Jo 2 (z) - qz.h2(Z), (' an n z 1 J i 2 (z)dz = 1 -Jo (z) - J 1 2 (z).

00J 1 2 (z)dz.

00If 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).

00(b2V2 + n2) (a2 - b 2) = - z It will now be convenient to introduce the quantities a l, a 2', 7731 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations Ty - 1 = dz ' 3= - dy 2 = dx - In terms of these we obtain from (7), by differentiation and subtraction, (b 2 v 2 + n 2) 7,3 = 0 (b 2 0 2 +n 2) .r i = dZ/dy (b 2 v 2 +n 2)', , 2 = - dZ/dx The first of equations (9) gives 3 = 0 (10) For al we have ?1= 47rb2, f dy e Y tkr dx dy dz

00Thus f (= 4-rb 2;JJ Z dY (e r) dx dy dz.

00These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation p = pk =RA (6) where 0 denotes the absolute temperature; and then d9 d p R dz - dz (p) n+ 1' so that the temperature-gradient deldz is constant, as in convective equilibrium in (I I).

00The 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.

00Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is - f'fpuq cos OdS = - f f (lpu 2 -+mpuv+npuw)dS, (1) which by Green's transformation is (d(uiu 2) dy dz dxdydz.

00The 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.

00These 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?

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.

00= dx dy dz the equations of motion may be Written du - 2v?

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.

00d o, dx dy dz dx dy dz so that, at any instant, the surfaces over which tk and m are constant intersect in the vortex lines.

00dz =o.

00Equation (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,.

00d - K dK dK _ dK dK dK ?dx n dyd °, udx dz - ° and K=fdp/o+V+2q 2 =H (3) is constant along a vortex line, and a stream line, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid.

00= 0, dz = O.

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+..

00u '= - 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?

00dt-(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'.

00d t dy dz where _ oo abcdA A, B ' C ' - (a 2 +A, b 2 ±x, A, c 2 +A) P P 2 = 4(a 2 -F-A) (b 2 ±A) (c2+A).

00Let 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).

00Let 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.

00Hence 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.

00Consider the integral W dx dy dz .

00Hence 8?rJJJ.dx l i 2 + i dy) 2 2 dxdydz= 8 Jfv d s_JJJvvvdx dngr dy dz.

00I by the whole area B"DZ'VO under the isothermal 9"D and the adiabatic DZ', bounded by the axes of pressure and volume.

00DZ, representing the loads taken in their order.

00These circuital relations, when expressed analytically, are then for a dielectric medium of types = (dt + x) (f',g',h')+dt(f,g,h), dR dQ = da dy dz dt' ' I See H.

00Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i d x +j - + k dz could be, like ix+jy+ kz, effectively expressed by a single letter.

00If x, y, z be the co-ordinates of P it is easily proved that dx ~ dz = V = di W =~ (1)

00~s Now, suppose a tracing point T Pa to be fixed to the cord, so as to be carried along the path of con- Dz a tact P11P2, that point will trace on a plane rotating along with the wheel I part of the involute of the base-circle DfD1, and on a plane rotating along with the wheel 2

00If we take the axis of z normal to either surface of the film, the radius of curvature of which we suppose to be very great compared with its thickness c, and if p is the density, and x the energy of unit of mass at depth z, then o- = f o dz, (16) and e = f a xpdz,.

00zpzL(z)dz, where, in general, we must suppose p a function of z.

00The surface-density of this stratum is a = cp. The energy per unit of area is e = f xpdz=cp(X' -4lrpe(o))+27rp'f c 0(z) dz+27rp fee(c - z)dz.

00Hence the surface-tension =e - =47rp 2 (f 0(z)dz - ce(c)).

00Integrating the first term within brackets by parts, it becomes - fo de Remembering that 0(o) is a finite quantity, and that Viz = - (z), we find T = 4 7rp f a, /.(z)dz (27) When c is greater than e this is equivalent to 2H in the equation of Laplace.

00If the density be a, the attraction between the whole of one side and a layer upon the other distant z from the plane and of thickness dz is 27r6 2 P(z)dz, reckoned per unit of area.

00The expression for the intrinsic pressure is thus simply K= 2 iro 2 f 1,G(z)dz (28) In Laplace's investigation o- is supposed to be unity.

00We may call the value which (28) then assumes Ko, so that as above Ko =27rf (z)dz.

00If a i, a 2 represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area 21ra l a fZ '(z)dz, (30) or 27ra l 02 0(z), if we write f 4,(z)dz=0(z) (31) The work required to produce the separation in question is thus 2 7ru l a o 0 (z)dz; (32) and for the tension of a liquid of density a we have T = a f o 0 (z)dz.

00For f0(z)dz =0(z).z - fz d dz) dz =0(z).z+fz4,(z) dz.

00Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply f 0(z)dz f: (z)dz, (34) and T= ref: z1,1,(z)dz (35) In Laplace's notation the second member of (34), multiplied by 27r, is represented by H.

00Integrating by parts, we get J l'(z)d z = zI, G (z) + 3 z 3 I I (z) 3 f z3Cb(z)dz, fzqi(z)dz = J z21 '(z) + k z41 I (z) + a fz4(1)(z)dz.

00In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write f o (z)dz = 3f2' z 3 4(z)dz, f o z(z)dz = 'a' z4 cb(z)d z; (36) so that Ko = 3 f o z3 ?(z) dz, To = $?

00o z 4 0(z)dz A few examples of these formulae will promote an intelligent comprehension of the subject.

00We may write T'12 = rQio f ° B(z) dz = bra i v 2 f 04 z 4 (z)dz;..

00Remember to do the three checks; Airspace, Alti and DZ, before starting any exercises and to respect all other canopy flyers.

00dz, by the hydranths, each with dactylozoid; gz, gastrozoid; b, mouth and tentacles; and, blastostyle; gon, gonophores; secondly, the " coenosarc," or rh, hydrorhiza.

00One class g g of polyps, the dactylozoids of branching in the Plumularia-type; (dz), lose their mouth and compare with fig.

00Thus so that z 1 J1 2 (z) = - 2 Jo 2 (z) - qz.h2(Z), (' an n z 1 J i 2 (z)dz = 1 -Jo (z) - J 1 2 (z).

00J 1 2 (z)dz.

00According 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).

00If 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).

00(b2V2 + n2) (a2 - b 2) = - z It will now be convenient to introduce the quantities a l, a 2', 7731 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations Ty - 1 = dz ' 3= - dy 2 = dx - In terms of these we obtain from (7), by differentiation and subtraction, (b 2 v 2 + n 2) 7,3 = 0 (b 2 0 2 +n 2) .r i = dZ/dy (b 2 v 2 +n 2)', , 2 = - dZ/dx The first of equations (9) gives 3 = 0 (10) For al we have ?1= 47rb2, f dy e Y tkr dx dy dz

00Thus f (= 4-rb 2;JJ Z dY (e r) dx dy dz.

00Phil.): - 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.

00These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation p = pk =RA (6) where 0 denotes the absolute temperature; and then d9 d p R dz - dz (p) n+ 1' so that the temperature-gradient deldz is constant, as in convective equilibrium in (I I).

00On 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.

00The 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.

00Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is - f'fpuq cos OdS = - f f (lpu 2 -+mpuv+npuw)dS, (1) which by Green's transformation is (d(uiu 2) dy dz dxdydz.

00The 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.

00These 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?

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 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.

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 ?

00= dx dy dz the equations of motion may be Written du - 2v?

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.

00d o, dx dy dz dx dy dz so that, at any instant, the surfaces over which tk and m are constant intersect in the vortex lines.

00dz =o.

00Equation (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,.

00d - K dK dK _ dK dK dK ?dx n dyd Ã‚°, udx dz - Ã‚° and K=fdp/o+V+2q 2 =H (3) is constant along a vortex line, and a stream line, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid.

00The 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?

00= 0, dz = O.

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(..

00u '= - 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?

00dt-(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'.

00d t dy dz where _ oo abcdA A, B ' C ' - (a 2 +A, b 2 ±x, A, c 2 +A) P P 2 = 4(a 2 -F-A) (b 2 ±A) (c2+A).

00Let 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).

00Let 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.

00Hence 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.

00Consider the integral W dx dy dz .

00We have by partial integration ff1 fV dd - ' 2 dy JJ dx y JJ y dxd dz = V - d dzdxd dz, and Itwo (similar equations in y and z.

00Hence 8?rJJJ.dx l i 2 + i dy) 2 2 dxdydz= 8 Jfv d s_JJJvvvdx dngr dy dz.

00I by the whole area B"DZ'VO under the isothermal 9"D and the adiabatic DZ', bounded by the axes of pressure and volume.

00DZ, representing the loads taken in their order.

00These circuital relations, when expressed analytically, are then for a dielectric medium of types = (dt + x) (f',g',h')+dt(f,g,h), dR dQ = da dy dz dt' ' I See H.

00Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i d x +j - + k dz could be, like ix+jy+ kz, effectively expressed by a single letter.

00The confusion of (3 with v necessitated the invention of a new symbol B in the Cyrillic, E in the Glagolitic for b, while new symbols were also required for the sounds or combinations of sounds z (zh), dz, Ã¯¿½t (sht), c (ts); c (ch in church), Ã¯¿½ (sh), u, i, y (u without protrusion of the lips), e (a close long e sound), for the combination of o, a and e with consonantal I (English y) and for the nasalized vowels e, q (nasalized o in pronunciation) and the combinations je and ja (English yg, ye).

00If x, y, z be the co-ordinates of P it is easily proved that dx ~ dz = V = di W =~ (1)

00~s Now, suppose a tracing point T Pa to be fixed to the cord, so as to be carried along the path of con- Dz a tact P11P2, that point will trace on a plane rotating along with the wheel I part of the involute of the base-circle DfD1, and on a plane rotating along with the wheel 2

00If we take the axis of z normal to either surface of the film, the radius of curvature of which we suppose to be very great compared with its thickness c, and if p is the density, and x the energy of unit of mass at depth z, then o- = f o dz, (16) and e = f a xpdz,.

00zpzL(z)dz, where, in general, we must suppose p a function of z.

00Hence if we write K =27rf o,/i(z)dz, H =27rf o zi/i(z)dz, the pressure of a column of the fluid itself terminating at the surface will be p2{K+1H(I/Rid-I/R2)}, and the work done by the attractive forces when a particle m is brought to the surface of the fluid from an infinite distance will be mp{K+zH(I/Ri+I/Ro)} If we write (.0 J then 27rmpo(z) will express the work done by the attractive forces, while a particle m is brought from an infinite distance to a distance z from the plane surface of a mass of the substance of density p and infinitely thick.

00The surface-density of this stratum is a = cp. The energy per unit of area is e = f xpdz=cp(X' -4lrpe(o))+27rp'f c 0(z) dz+27rp fee(c - z)dz.

00Hence the surface-tension =e - =47rp 2 (f 0(z)dz - ce(c)).

00Integrating the first term within brackets by parts, it becomes - fo de Remembering that 0(o) is a finite quantity, and that Viz = - (z), we find T = 4 7rp f a, /.(z)dz (27) When c is greater than e this is equivalent to 2H in the equation of Laplace.

00We have given several examples in which the density is assumed to be uniform, because Poisson has asserted that capillary B (25) e = c p (X' - 4 7rpe (0) ) -I-474Ã‚°o(z)dz.

00If the density be a, the attraction between the whole of one side and a layer upon the other distant z from the plane and of thickness dz is 27r6 2 P(z)dz, reckoned per unit of area.

00The expression for the intrinsic pressure is thus simply K= 2 iro 2 f 1,G(z)dz (28) In Laplace's investigation o- is supposed to be unity.

00We may call the value which (28) then assumes Ko, so that as above Ko =27rf (z)dz.

00If a i, a 2 represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area 21ra l a fZ '(z)dz, (30) or 27ra l 02 0(z), if we write f 4,(z)dz=0(z) (31) The work required to produce the separation in question is thus 2 7ru l a o 0 (z)dz; (32) and for the tension of a liquid of density a we have T = a f o 0 (z)dz.

00For f0(z)dz =0(z).z - fz d dz) dz =0(z).z+fz4,(z) dz.

00Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply f 0(z)dz f: (z)dz, (34) and T= ref: z1,1,(z)dz (35) In Laplace's notation the second member of (34), multiplied by 27r, is represented by H.

00Integrating by parts, we get J l'(z)d z = zI, G (z) + 3 z 3 I I (z) 3 f z3Cb(z)dz, fzqi(z)dz = J z21 '(z) + k z41 I (z) + a fz4(1)(z)dz.

00In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write f o (z)dz = 3f2' z 3 4(z)dz, f o z(z)dz = 'a' z4 cb(z)d z; (36) so that Ko = 3 f o z3 ?(z) dz, To = $?

00o z 4 0(z)dz A few examples of these formulae will promote an intelligent comprehension of the subject.

004,(z) =0 from z = a to z=co, Equations (37) now give 2 it " ira4 Ko = - z i dz = 6, 3 0 _ 7r fa 4 sra5 T o 8 fo zdz=40 The numerical results differ from those of Young, who finds that " the contractile force is one-third of the whole cohesive force of a stratum of particles, equal in thickness to the interval to which the primitive equable cohesion extends," viz.

00We may write T'12 = rQio f Ã‚° B(z) dz = bra i v 2 f 04 z 4 (z)dz;..

00dz, by the hydranths, each with dactylozoid; gz, gastrozoid; b, mouth and tentacles; and, blastostyle; gon, gonophores; secondly, the " coenosarc," or rh, hydrorhiza.

01One class g g of polyps, the dactylozoids of branching in the Plumularia-type; (dz), lose their mouth and compare with fig.

01According 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).

01We have by partial integration ff1 fV dd - ' 2 dy JJ dx y JJ y dxd dz = V - d dzdxd dz, and Itwo (similar equations in y and z.

01(11), where r is the distance between the element dx dy dz and the point where a l is estimated, and k = n/b = 27r/X (12), X being the wave-length.

02(11), where r is the distance between the element dx dy dz and the point where a l is estimated, and k = n/b = 27r/X (12), X being the wave-length.

02

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