JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Theoretical Interpretation of Moir6 Patterns

GERALD OSTER, MARK WASSERMAN, AND CRAIG ZWERLINGDepartmnent of Chemnistry, Polytechnic Institute of Brooklyn, Brooklyn, New York 11201

(Received 26 September 1963)

A general mathematical technique for the solution of moir6 patterns produced by the overlapping of twofigures is presented. The technique is applied to combinations of figures involving parallel lines, radial lines,and concentric circles including those in which the spacing is variable. When an equispaced parallel linefigure is overlapped on a parallel line figure whose spacing is variable, the resultant moire pattern revealsthe functional form of the variation.

The theory of the measurement of refractive index gradients by the moire technique is presented. Threearrangements of the positions of the figures with respect to the sample are analyzed. One compact arrange-ment gives exclusively the refractive index gradient. The interpretation of moir6 patterns distorted by lensesis considered in terms of the properties of the lenses.

The mathematical solutions of moir6 patterns are, in many cases, identical with those arising in physicalproblems. Examples are given for a number of phenomena arising in physical optics, hydrodynamics, andelectrostatics.

INTRODUCTION

THE first to explain how moir6 patterns are pro-Tduced from the near superposition of two familiesof equispaced parallel straight lines was Rayleigh.' Inmore recent years the moir6 technique for this case hasbeen exploited for the evaluation of replica gratings2 andfor metrology generally. 3 One of us (G.O.) in collabora-tion with Y. Nishijima4 demonstrated that interestingmoir6 patterns could be obtained with two families ofcircular figures and with circular figures and straightlines. In the present work a mathematical analysis ofsuch systems as well as that for other systems, includingfigures consisting of radial lines, is given. Many of thesesystems are analogs of physical systems including thoseencountered in certain areas of physical optics, hydro-dynamics, and electrostatics. The last section of thepresent paper is concerned with such questions.

It has been demonstrated that refractive indexgradients can be measured by means of the moiretechnique employing two gratings of equispaced parallellines.5 This sensitive method requires only simple andcompact apparatus. It could, conceivably, supplantother refractive index methods, namely, the Lammscale, schlieren, and interference methods, 6 which arecurrently employed for the determination of refractiveindex gradients in ultracentrifuge, diffusion, and elec-trophoresis apparatus. For this reason we give below adetailed analysis of the moir6 patterns produced bythe overlapping of a figure consisting of parallel equi-spaced lines onto that with variable distances betweenthe lines.

1 Lord Rayleigh (J. W. Strutt), Phil. Mag. 47, 81, 193 (1874).2 J. Guild, The Interference Systems of Crossed Diffraction

Gratings (Oxford University Press, London, 1956).J. Guild, Diffraction Gratings as Measuring Scales (Oxford

University Press, London, 1960).G. Oster and Y. Nishijima, Sci. Am. 208, May, 54-63 (1963).V. Nishijima and G. Oster, J. Opt. Soc. Am. 54, 1 (1964).For review with extensive bibliography, see H. Svenson in

Analytical Met/hods of Protein Clhemistry, edited by P. Alexanderand R. J. Block (Pergamon Press, Ltd., London, 1961), Vol. 3.

INDICIAL REPRESENTATION OF FIGURESAND PATTERNS

A moir6 pattern may be described as a locus of pointsof intersection of two overlapping figures. If each of thetwo original figures is regarded as an indexed family oflines, the resulting moir6 pattern is most pronouncedwhen the indices of the intersections satisfy certainsimple relations. Let F(xy) =V(h) represent the firstindexed family of curves and G (x,y)= cJ(k) representthe second indexed family where h and k are the index-ing parameters running over some subset of the realintegers and x and y are the coordinates of any pointon the plane of the figure. The functions F(x,y) andG(x,y) determine the form of the elements of the figuresand 41(h) and c)(k) determine the spacing of the ele-ments, an element being one of the lines of a figure.Thus the resultant moire pattern is an indexed familyof curves whose index p satisfies the indicial equation0(h,k)=p where p runs over some subset of the realintegers. For many of the two-figure systems consideredin the present work the indicial equation is givensimply by

h-=p(1)i.e., e) (hk) = h-k. We note any cases in which a differentindicial equation is appropriate to describe the re-sultant moir6 pattern.

A number of examples of the indicial representationmethod for the determination of the moir6 pattern ob-tained by overlapping two figures is now given. Thesimplest case, namely that for two sets of equispacedparallel lines, can, of course, be solved by elementarygeometry1' 5 or by a vectorial argument7 but for othercases our general indicial method avoids some verycumbersome mathematics. It is instructive to considerin detail the simplest case by the indicial method sincethis illustrates certain aspects of the mathematicaltechnique.

7 G. L. Rogers, Proc. Phys. Soc. (London) 73, 142 (1959).

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VOLUME 54, NUMBER 2 FEBRUARY 1964

o1s rSIER, WASSERMAN, AND ZWERLTNG

. . . -3 -2 -I. -. &p to the dotted line in Fig. 1, but they are less prominent- due to the lower density of points of intersection along

the fringe. These other moire patterns are represented.- 0 by the indicial equations

h-rk= p(r-4 1),

where r is any rational number. Here the distancebetween the fringes and the sine of the angle which

° the fringes make with the x axis is also given by Eqs. (6)and (7), respectively, but now b is replaced by rb.

Case II. Equispaced Concentric Circles onEquispaced Parallel Lines

- 2

... -2 -I 0 1 2 3 4 S -/

icG. 1. Case I.

Case I. Equispaced Parallel Lines onEquispaced Parallel Lines

The spacing of the elements of the first figure is b andof the second figure in a (see Fig. 1). The indices of thefirst figure are given by /z=0, -1, -2, h3, * -- and ofthe second by k=0, 4-1, 4t2, 43 * -. As seen fromFig. 1, the equation for first family of curves is

x= b/, (2)

i.e., F(x,y)=.x/b and Vp(h)= It, and for the secondfamily is

y= x cot6- (ak/sinO), (3)

The equations for the figures are given by

x2+y2= (ha)2 (h>O)and

x=bk (-oc<k<+co),

where a and b are the intercircle and interline spacings,respectively. The moire patterns occur for the indicialequation

it-k=ip, (11)

where p>0 if a>b and p<0 if a<b. Eliminating h andk from Eqs. (9)-(11), we obtain for the equation of themoire patterns

(b2-a

2)x

24 2a 2 bpx+b 2y2= a 2b2

p2.

This equation represents hyperbolas, ellipses, and para-bolas for a>b, a~b, and a=b, respectively.8 Thisillustrates the variety of moire patterns which can beobtained by varying the relative spacings of the twofigures.

i.e., G(xy)= (x cosO-y sinO)/a and cD(k)=k. The mostprominent moir6 fringes (the dashed lines in Fig. 1)correspond in this case to the indicial equation, Eq. (1).

Eliminating It and k from Eqs. (1)-(3), we obtain asthe equation of the moir6 pattern

/b cos0-a\ paY= X1 (4)

b sinO sinO

which may also be written as

y = x cot p- (pd/sin~p), (5)

where d is the distance between fringes and so is theangle which the fringes make with the y axis. Equatingcorresponding coefficients of Eqs. (4) and (5), we obtain

d= ab/ (a2 +b 2-2ab cosO)i (6)

andsinp= b sinO/(a2 +b2 -2ab cosO)', * (7)

in agreement with the solution obtained by elementarygeometrical reasoning.'

There are other moire patterns such as those parallel

Case III. Equispaced Concentric Circles onEquispaced Concentric Circles

When the centers of the circles coincide, beats areobserved at distances from the center given by pab/a - bwhere a and b are the intercircle spacings in the twofigures and p is a positive integer. For the trivial casewhen the two figures are identical and are exactlysuperposed the beat occurs, of course, at infinity.When the two identical figures are displaced withrespect to one another, however, a moire pattern con-sisting of hyperbolas with the centers of the circles asfoci is produced. 5 Still further, ellipses occur with thecenters as foci. This is particularly evident to the eyewhen the distance between centers is a few times greaterthan the intercircle spacing (Fig. 2). The equationsfor the two figures are

and(x-s) 2 +-l-y2 h= 2a2

(X+s) 2 +y 2= k 2a2,

(13)

(14)

8 G. Edgecombe (personal communication) obtained the sameresult independently.

(8)

(9)

(10)

(12)

1.70 Vol. 54

February1964 THEORETICAL INTERPRETATION OF MOIRE PATTERNS

where the distance between centers is 2s. The indicialequation is given by

h?-k = p. (15)

By elimination of h and k from these equations oneobtains

x2 y2

i =1.p2 /4 p2 /4-S 2

When the negative sign is chosen, hyperbolas are ob-tained and the positive sign gives ellipses.

Case IV. Fresnel Figure on AnotherFresnel Figure

By "Fresnel figure" we mean the standard positiveFresnel zone plate consisting of concentric circles alter-nately blackened out and constructed in such a waythat the areas of the black and the nonblack portionsare equal.9 The equations of the figures are given by(the intercenter distance being 2s)

(X- S) 2+y 2 = h/wr (17)

and(X+S)

2+y

2== kw, (18)

taking each of the areas as equal to unity. The indicialequation for the moir6 pattern is given by Eq. (1).Eliminating h and k from Eqs. (17), (18), and (1), weobtain for the equation of the resulting moire pattern

x==p127r(2s).

FIG. 3. Case VI.

Case V. Fresnel Figure on EquispacedParallel Lines

This case is particularly interesting because the moir6pattern consists of many zone plates, i.e., multiplereplications of the original figure occur.4 The equationsfor the two figures are given by

X2+ y2= 1/7r, (20)

(19) x = ka. (21)

That is, the moir6 pattern consists of parallel straightlines (see Ref. 4) whose spacing is d= (47rs)-l. Inci-dentally, the appearance of this simple moir6 patternprovides a criterion for evaluating the accuracy of zoneplates.

To account for multiple moir6 patterns the indicialequation is written as

h-rk=p, (22)

where r==0, 4z1, 42, 3 - - -.Eliminating h and k from Eqs. (20)-(22) gives the

equation for the moire pattern

(23)

FIG. 2. Case III.

I See, for example, M. Born and E. Wolf, Principles of Optics(Pergamon Press, Ltd., London, 1959), Sec. 8.2.

This equation is, for a given value of r, merely theequation for a single Fresnel figure displaced a distancer/27ra from the original origin. Furthermore, the phaseof the original figure has now been shifted in the moir6figure by a value of (r/27ra)2 . Hence for given Fresnelfigure in the moir6 pattern (i.e., a given value of r) aphase shift of a half period (i.e., black to white) willoccur if the spacing of the straight line figure a= r/ (27r).

Case VI. Radial Lines on Radial Lines

For simplicity the radial lines are taken to be equi-angular (see Fig. 3). The equations for two such figures

(16)

171

2 P r 2X_ +�2 = _+

7 27ra

1OSTrER, WASSERMAN, AND ZWERLING

FIIG. 4. Case VII.

whose centers are a distance 2s apart are given by

y= (x+s) tan(k7r/c), (24)

y= (x-s) tan(Ih7r/c), (25)

where c is the number of lines in the figure. Moir6patterns correspond to the indicial equation, Eq.(1). Eliminating It and k from Eqs. (24), (25), and(1), using the relation that tan(oz+3)= (tana+tan3)/(1-tana tan3), one obtains for the moire pattern

X2+ Q ) [ Q (26)

where Q=tan(p7r/c). That is, the pattern consists ofcircles of radii given by the bracketed term in Eq. (26)whose centers all lie on the y axis. Furthermore, eachof these circles passes through the two points (s,0) and(-s, 0). These circles are more easily observed whenalternate sectors of the radial figures are blackened out.

Case VII. Radial Lines on EquispacedParallel Lines

This case is similar to Case V in that multiple replica-tions occur (Fig. 4). The equations for the two figuresare given by

y=x tan (i7r/c) (27)

andy=ka. (28)

In conformity with this equation, the moire pattern inFig. 4 has a twofold symmetry about the x axis andabout the y axis. The larger the spacing in the parallelline figure, the larger will be the scale of the pattern.For I r I > 1 multiple replications of the pattern shownin Fig. 5 are observable wherein the scale factor of themultiple patterns increases with increasing value of I x

PARALLEL LINES OF VARIABLE SPACINGAND THEIR APPLICATIONS

In the moire technique for the measurement of re-fractive index gradients5 it is necessary to relate theobserved moir6 pattern to the actual spatial distribu-tion of the gradient. The moire pattern produced is thatfor the overlapping of an equispaced parallel line figurewith one where the spacings follow a certain functionwhich is determined by the distortions optically pro-duced by the gradient.

Consider a family of lines x= ah modified by a func-tion f(x). The equation for this new figure is given by

x= bh+tf (bh), (30)

where I is a constant. Now overlap on this figure afamily of equidistant parallel lines of spacing b/cosOwhere 0 is the angles through which this figure is rotatedclockwise. Then Eq. (3) becomes

y= x cotO- (bk/sinO cosO). (31)

The produced moire pattern corresponds to the indicialequation Eq. (1) in all cases in which the sequencex(h), Eq. (30), is monotonic, that is, x(h)>x(h") if andonly if It'> h". To satisfy this condition the functionf(x) must be itself a monotonic function or it must bebounded. In the case that the function is bounded butnot monotonic, e.g., the Gaussian error curve, a con-stant t in Eq. (31) can always be chosen small enoughso that x(h) is an order-preserving sequence. Eliminatingh and k from Eqs. (30), (31), and (1), we obtain for theequation of the moire pattern

X=w±tf(w), (32)

where w= cos6(x cos6-y sin0)+bp. This equation issimplified if it is referred to a coordinate system formedby a clockwise rotation of angle 0 with respect to theoriginal coordinate system. In this new (primed) coordi-nate system Eq. (32) becomes

y'= bp sec6+t sec~f(x' cos6+bp). (33)

The moir6 pattern corresponds to the indicial equa-tion, Eq. (22). On elimination of At and k from Eqs. (27),(28), and (22), one obtains for the equation of themoire pattern

C y ry- tan-'---= p. (29)7r x a

The introduction of the 1/cosO factor in Eq. (31)eliminates an extraneous x' cosO term.

As an example of this general result consider the caseof a figure of parallel lines whose spacing b is modifiedby a Gaussian function. Then the equation for thisfigure is

x= ah+exp- (ah)2 , (34)

172 Vol. 54

February1964 THEORETICAL INTERPRETATION OF MOIRE PATTERNS

where 1= 1 since we have taken a> 1. This assures thatx(h) is monotonic. The equispaced parallel line figure isdescribed by Eq. (3). Eliminating h and k from Eqs.(34), (31), and (1), yields for the equation of theresulting moire pattern

x= av+exp- (av)2, (35)

where v= (x cos6-y sinG)/b-p. Referring this to theprimed coordinate system above and setting b = a/cos6,Eq. (33) yields the much simplified result

y'= secO exp- (x' cosO-pa) 2]-secOpa. (36)

Hence the resultant moire is a collection of identicalGaussian curves whose position is determined by theindex p and whose form is precisely that of the originalmodifying function.

The bending of light by a refractive index gradientwas first treated by Wiener10 and the theory was utilizedby Lamm" who considered the distortion producedwhen a parallel line figure is viewed through a gradient.The theory has been further refined by Svensson.'2 Forsimplicity we assume that the illumination consists ofa parallel beam falling at right angles to the parallel-walled cell containing the gradient which is in the xdirection (Fig. 5). The displacement x,-xo of the rayemergent at Position 2 can be represented as a Taylorseries which for incident light at right angles to thecell (of thickness A) reduces to

A2 1 do\2 n dx/ x=xO

when the region in the cell swept by the ray is so smallthat the gradient can be regarded as essentially con-stant within it.'2 The displacement Z at Position 3 isgiven by"'

Z = A C (dnldx) _x° (38)

Consider now three different experimental arrange-ments (Fig. 5). Throughout the line figure correspond-ing to Eq. (2) is placed on the left side of the alteredequispaced parallel line figure corresponding to Eq. (31).In arrangement a the figure at Position 1 appears onemergence from the cell at Position 2 as the distortedimage described by the equation

A 2 1 dnx= bh+-( - (39)

2 \n dx=bh

Hence from the Eq. (33) the resultant moire pattern ob-served when the altered equispaced parallel line figureis placed at Position 2, is described by the equation

A 2 1 duy'= bp secO+-sec6- a(40)

2 n d(x' cosd+bp)

0. Wiener, Ann. Physik. 49, 105 (1893).01 0. Lamm, in T. Svedberg and K. 0. Pedersen, The Ultracen-

trifuge (Oxford University Press, London, 1939).12 H. Svensson, Opt. Acta 3, 164 (1956).

e-A C >

%, %-----------

I 234 3FIG. 5. Schematic arrangements for refractive index gradient

measurements. The y coordinate is perpendicular to the figure.Arrangements a, ,3, and -y correspond to parallel line figures at1 and 2, 1 and 3, and 2 and 3, respectively. (Note: angles in thediagram are exaggerated.)

where the refractive index gradient is sufficiently smalland of a form to maintain order of the lines in the dis-torted image. Hence one observes a moire pattern con-sisting of series of curves which correctly representn' (dnjdx) as function of distance x along the cell.

If, however, the altered figure is placed at Position 3(the A arrangement) one must consider the refractionof the ray at the cell-air interface (Position 2). Thenthe displacement of a line at Position 3 is Z± (xa,-xo).This becomes approximately Z [Eq. (38)] for smalldisplacements, relative to Z, of the ray at Position 2,that is, for C>>A. Under these conditions the moirepattern is described by the equation

dny'=bp secO+CA secO d coso-bp) (41)

that is, the x variation of the gradient alone isdetermined.

One avoids the necessity of the approximation C>>Aby simply placing the figures at Positions 2 and 3 (the-y arrangement). In this way we observe only the linedisplacements formed between Positions 2 and 3 dueto the bending of the light in the cell. Hence C need notbe large compared to A since there is now no necessityto minimize the displacement xa-xo relative to Z. Bycombining the results of experiments using the a and the-y arrangements, one obtains the dependence of the re-fractive index on distance (x direction) in the cell.

The moir6 technique can be utilized to characterizelenses.4 " When the image of a figure produced by a lensis overlapped with an identical figure the moir6 patternof the image differs from that of the background. Thesedifferences may be related to the focal length and toaberrations of the lens. Suppose the first figure con-sisting of equispaced parallel lines (spacing c) is illumi-nated with a beam parallel to the principal axis of thelens. If the second identical figure is placed at a distances from the center of a thin perfect lens of focal lengthf(sKf) then the image at s has a spacing c(f-s)lf.Hence the spacing and sine of the angle of the moirepattern at the image are given by Eqs. (6) and (7)

173

I0

TZ

OSTER, WASSERMAN, AND ZWERLING

where a is replaced by c and b is replaced by c(f-s)/f.Solving for f we obtain in terms of q, the spacing of theoriginal figure relative to the distance between fringesin the moire pattern, i.e., q= bd:

(1-q2 )sf= q2)S(42)

1- cosO-q2 T (q2-sin20)2

and in terms of the angle p of the fringes:

f=s sin(soiS)/Esin(soiS)-sinso], (43)

where the positive sign refers to a convergent lens and anegative sign refers to a divergent lens. Hence, the focallength is determined either by the spacing of the fringesin the image of the lens or, more simply, by the rotationof the fringes. Variations in focal length within the lens(i.e., aberrations) are manifested as curved moir6fringes. The variation in focal length Af at any pointin the lens is obtained by differentiating either Eq. (42)or Eq. (43). Taking the more tractable expression,Eq. (43), one obtains for the relative change in focallength

Afcot <p-cot ( (pi) A\ ((44)

f sin( pozt0)-sin sinsp

Zimmerman1 3 has measured the spacings of the imageof a line ruling by means of the moire technique andrelates variations in spacing to aberrations in the lens.

PHYSICAL INTERPRETATION

There are a number of physical problems whosemathematical solution is also the solution of a moir6problem. In this sense moire patterns are analogs ofthe physical phenomenon in question.

The analogy of diffraction of light by a pile of thinplates (equivalent to x-ray diffraction by a crystallattice) to Case I (where a=b=X, the wavelength oflight, and d is the interplanar spacing) has alreadybeen pointed out.5 Case II would correspond to theinterference between coherent light beams, one beingspherical and the other planar. Case III should corre-spond to the interference between two coherentspherical waves (i.e., Young's experiment) but inactual fact only one class of moir6 figures has physicalmeaning. The requirement for interference in thephysical situation is that it occurs when the differ-ence of the distances from the centers of the figures isconstant, that is, that the points of interference be alonghyperbolas. This condition is met, as we have seen, ifin the indicial equation, Eq. (15), the negative sign ischosen. On the other hand, if the sign is positive themnoirel descrilbes ellipses, but this is devoid of physicalmeaning.

The moire technique can be utilized to solve certain

13 J. Zimmermann, Appl. Opt. 2, 759 (1963).

problems in potential theory. Figures corresponding tothe equipotentials about electrical charges can be repre-sented as concentric circles whose radii vary inverselyas integers. The overlapping of two such figures (theirrelative size determined by the relative magnitude ofthe charges) results in a moir6 pattern correspondingto the equipotentials for an electric dipole. The re-sultant moir6 corresponds to the indicial equation,Eq. (1). Since h and k are proportional to the potentialsand potentials can be added as scalars, all pointssatisfying this equation for a given value of p have thesame potential which is proportional to p. The negativesign indicates that the potentials have opposite signand hence describes a configuration for opposite charges,i.e., a dipole. On the other hand, for h+k=p, namelythe case of charges of the same sign, the figures do notyield a readily perceptible moire pattern. Using theindicial equation, Eq. (15), and the equations for thefigures, namely,

(x+a) 2+y 2 = (blk)2,

(x-a) 2 +y 2 = (C/h)2,

(45)

(46)

where b/c is determined by the charge ratio, then theresulting moir6 pattern is

b c

(x+a)2+y2 (X-a) 2+y2 (47)

where p is proportional to the potential of the moirefringe. The negative sign gives the equation for the mostpronounced moire, namely, that for a dipole.

In an analogous manner the equations of the moirepatterns corresponding to other pairs of charge con-figurations can be derived. Although the intersectionsof the figures will lie on the equipotentials of the re-sulting configuration it does not follow that the moir6pattern will be particularly prominent.

The motions due to a two-dimensional uniformstream and any number of two-dimensional sources (orsinks) can be obtained mathematically by addition ofthe corresponding complex potentials.'4 A hydrody-namic source in a uniform stream is represented by CaseVII where p is equivalent to the stream function andthe spacing of the parallel lines is inversely proportionalto the velocity of the uniform flow field. The indices Itand k represent the stream functions of a source and auniform flow, respectively. By holding the stream func-tions constant (setting It and k equal to integers) wedescribe the streamlines since they are defined as linesalong which the stream function has a constant value.The resultant stream function can be expressed as thesum of the two original stream functions'4 and hence theintersections lying on the pth moir6 fringe form astreamline of the resulting configuration. By setting

14 L. M. Milne-Thomson, Tlieoretical Hydrodynamics (Mac-millan and Company, Ltd., London, 1950), 3rd ed., Chaps. 4 and 8.

174 Vol. 54

Februaryl964 THEORETICAL INTERPRETATION OF MOIRE PATTERNS

our stream junctions equal to multiples of unity, thestreamlines become the boundaries of flux tubes. Thispoint is pursued somewhat further in the discussion ofthree-dimensional phenomena. At any rate, our moirepattern is symmetrical, unlike the actual hydrodynamiccase since the lines representing the uniform flow,namely the parallel straight lines, have no directionalcharacter.

A radial line figure corresponds to the flow field of ahydrodynamic source in a plane, the lines representingthe streamlines. The overlapping of two such figures(Case VI) yields a moire pattern which describes theflow field for a hydrodynamic dipole consisting of asource and sink of equal strengths. Similarly we couldgeneralize the problem for sources of unequal strength,the strength being determined by the parameter c. Ourfigures do not indicate the direction of flow, that is,whether we are dealing with a source or a sink. Sincethe prominent moir6 is governed by the indicial equa-tion, Eq. (1), the observed moire pattern correspondsto a subtraction of the contributing stream functions,in other words, to a source-sink combination.

When one is concerned with a physical situationwhich is inherently three-dimensional in nature, it isnecessary for the purposes of the moire technique toconstruct figures which are two-dimensional "perspec-tives." This is the case, for example, when dealing withthe interaction of the fields (lines of force) in the regionof two' point charges. Inasmuch as the index of eachfigure increases by unity from element to element,these elements which are lines of force must of necessitybe the boundaries of a collection of three-dimensional

unit flux tubes. As a consequence," the equations forthe two figures corresponding to two charges separatedby a distance 2s are

(48)

(49)

y= (x-s){[1+ (hdi)'},f/IdJ},y== (x+s){[l+ (hdQ)]112hd2),

where dl/d2 is the inverse ratio of the correspondingcharges. The spacing d is the uniform distance betweenprojection onto the x axis (the axis connecting thecharges) of the intersections of the radial lines with aunit circle. These equations together with the indicialequation Eq. (15) yield for the equation of the moir6pattern when di =d2=d

(x-s) (x+s)± i = pd.Ey2+ (X- S)2 If Ey2+ (XS+)2]l~

(50)

The equation for the prominent moir6 pattern is thatobtained when the minus sign is chosen, i.e., the caseof the three-dimensional dipole. Some textbooks onelectricity' in describing a construction of the dipolefield make the error of representing the field of a pointcharge not as a "perspective" but rather as a crosssection. As a result they obtain the streamlines of ahydrodynamic dipole in two dimensions instead ofthe lines of force of an electrostatic dipole in threedimensions.

1' C.f., J. C. Maxwell, A Treatise on Electricity and Magnetism,3rd edition, 1891 (reprinted by Dover Publications, Inc., NewYork, 1954), pp. 177-185.

16 E.g., R. W. Pohl, Einfidirutng in die Elektrizitdtslelhre (JuliusSpringer-Verlag, Berlin, 1927), p. 61, Fig. 112.

175