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the gravitational lensing effect of an ensemble of point masses has long been a subject of investigation ( see , e.g. , schneider , ehlers , & falco 1992 for a summary of earlier work ) . most of the studies have focused on the situation where the spatial distribution of the point mass lenses is `` compact , '' i.e. , the size of the region within which the lenses are distributed is much smaller than the other distances ( those between the source and the lenses and between the observer and the lenses ) in the problem . for gravitational lensing calculations , a compact distribution of lenses is usually approximated by its projection onto a single lens plane . the important case of low optical depth , single plane microlensing has been investigated in great detail ( kaiser 1992 ; mao 1992 ; rauch  1992 ; kofman  1996 , hereafter paper i ; see also schneider  1992 and references therein ) , and we now understand reasonably well the caustic structure , the illumination pattern , the probability distribution of macroimage magnification , and the flux variability of macroimages in this case .    there are , however , situations in which the lensing objects are modeled by an `` extended '' ( or three - dimensional ) distribution of point masses , i.e. , by a point mass distribution that is comparable in scale to the other distances in the problem . a well - known example is the microlensing of stars in the large magellanic cloud and the galactic bulge by foreground stars and possibly massive compact halo objects ( machos ) ( alcock  1993 ; aubourg  1993 ; udalski  1993 ) . the theoretical analysis of this problem is relatively simple because the optical depth to microlensing is very small ( @xmath9 ) and the point mass lenses are essentially acting independently ( paczyski 1986 , 1991 ; griest 1991 ) . ( note , however , that microlensing by physical binary systems is also possible ; mao & paczyski 1991 ; udalski  1994 ; alard , mao , & guibert 1995 . ) a more complicated problem is the possibility that a substantial fraction of the dark matter in the universe is in the form of compact objects ( e.g. , carr 1994 ) . then the universe as a whole has a significant optical depth to microlensing ( press & gunn 1973 ; dalcanton  1994 ) . this latter problem has been studied by several authors using numerical simulations ( refsdal 1970 ; schneider & weiss 1988a , b ; rauch 1991 ) , and it was found that there are qualitative differences between two- and three - dimensional microlensing . for example , for @xmath5 , the three - dimensional macroimage magnification distribution does not show the prominent caustic - induced feature produced by a planar distribution of point masses ( rauch  1992 ; but see   4 below ) . the usual approach to gravitational lensing by a three - dimensional distribution of lenses is the multiple lens - plane approximation ( blandford & narayan 1986 ; kovner 1987 ) . the properties of multiplane gravitational lensing and , in particular , the reasons for many of the differences between two- and three - dimensional microlensing have not been investigated in detail , and they are not yet well understood . besides the statistical microlensing studies mentioned above , there have been few studies of the basic mathematical properties of multiplane gravitational lensing ( levine & petters 1993 ; levine , petters , & wambsganss 1993 ) or the properties of specific multiplane configurations ( kochanek & apostolakis 1988 ; lee & paczyski 1990 ; erdl & schneider 1993 , hereafter es ) . this was perhaps due to the lack of lensing candidates with lenses at different redshifts . however , recent observations of the lens systems 2345@xmath10007 ( pell 1996 ) , 2016@xmath10112 ( lawrence , neugebauer , & matthews 1993 ) , and 0957@xmath10561 ( bernstein , tyson , & kochanek 1993 ; but see dahle , maddox , & lilje 1994 ) suggest that these systems may involve lenses at different redshifts .    as we shall see in   3 , to understand some of the differences between two- and three - dimensional microlensing , it is useful to study in detail specific cases of simple three - dimensional lens configurations the study by erdl & schneider ( es ) of the lens system consisting of two point masses at different distances from the observer is a step in this direction . they considered primarily the critical curves and the caustics and were able to obtain a complete classification of their topological properties with respect to the model parameters in this particular case . the main motivation of this study is to give a systematic theory of gravitational microlensing by a random distribution of point masses . in the accompanying paper i we refined the theory of gravitational microlensing for a planar distribution of point masses . in particular , we obtained a practical semi - analytic expression for the macroimage magnification distribution , @xmath4 , for low optical depth @xmath11 , by allowing for the shear perturbations of neighboring lenses . in this paper we study the properties of gravitational microlensing by a three - dimensional distribution of point masses . we are interested in the critical curves , the caustics , the illumination pattern , and especially the macroimage magnification distribution . we shall not consider other interesting aspects such as time delays and the temporal behavior of microlensing events ( note , however , that the latter is given by the magnification along tracks through the illumination pattern ) . in 2 we write down the lens equations . in   3 we study in detail multiplane configurations with two , three , and four point masses to find out what new features are expected in three - dimensional microlensing . we also explain qualitatively why these new features arise . the case of low optical depth three - dimensional microlensing is studied in 4 , using an approach similar to that used in paper i to study the caustic - induced feature in @xmath4 for the planar case . the resulting semi - analytic @xmath4 is compared to the numerical results of rauch ( 1991 ) , and the differences between the two- and three - dimensional results are discussed . our conclusions are summarized in we begin by establishing the relevant equations for gravitational lensing by a three - dimensional lens distribution . as in paper i , we shall only consider gravitational lensing in an einstein - de sitter cosmological background weakly perturbed by the gravitational field of the lenses . our notation is identical to that used in paper i , and the readers are referred to   2 of paper i for the definitions of terms not explicitly defined here . let us consider an isotropic point source at the origin of our coordinates and an observer plane at a comoving distance @xmath12 from the source ( see fig .  1 ) . in the absence of perturbations , the observer plane would be uniformly illuminated and a light ray which leaves the source with angle @xmath13 would pierce the observer plane at @xmath14 . this defines our planar lagrangian coordinates @xmath15 . in the presence of density inhomogeneities , the light ray would suffer deflections and pierce the observer plane at the eulerian coordinates @xmath16 . for a three - dimensional ( or extended ) lens distribution , gravitational lensing is usually studied within the framework of the multiple lens - plane approximation ( blandford & narayan 1986 ; kovner 1987 ) . it is assumed that the lens distribution can be approximated by a series of @xmath17 geometrically thin configurations , i.e. , lens planes , and that the separations between the lens planes are sufficiently large . then the deflection of a light ray by one of the lens planes is not influenced by the others and a light ray is deflected @xmath17 times on the way from the source to the observer ( see fig .  1 ) . after @xmath18 deflections , the location at which the light ray pierces the @xmath19th lens plane , projected onto the observer plane , is @xmath20 where @xmath21 is the deflection by the @xmath22th plane ( which can be expressed as the gradient with respect to @xmath23 of an effective surface gravitational potential on plane @xmath22 ) , @xmath24 is a dimensionless distance parameter defined by @xmath25 and @xmath26 is the comoving distance between planes @xmath22 and @xmath19 . note that @xmath24 can take on values between 0 ( in the limit @xmath27 ) and 1 ( in the limit @xmath28 or @xmath29 ) . if we denote the observer plane as plane @xmath30 , then the lagrangian to eulerian mapping is    @xmath31    where @xmath32    unlike the single plane lens mapping , this multiplane lens mapping is not in general a gradient mapping .    for the particular case where the lens planes are populated by point masses , @xmath33 where @xmath34 is the cosmological scale factor at the redshift of lens plane @xmath22 and @xmath35 is the location of the @xmath36th lens ( of mass @xmath37 ) on lens plane @xmath22 . note that @xmath35 is the position projected onto the observer plane , i.e. , @xmath38 where @xmath39 is the angular position of the lens from the source . henceforth , we shall assume for simplicity that all the point masses have the _ same _ mass @xmath40 . then @xmath41 where @xmath42 is the ( lagrangian ) einstein radius of a point mass on the @xmath22th lens plane . for a random distribution of point masses , the optical depth to microlensing is @xmath43 , where @xmath44 and @xmath45 are , respectively , the projected ( onto the observer plane ) surface number density and the physical surface mass density of point masses on lens plane @xmath22 . when we consider the situation where a significant fraction of the dark matter in the universe is in the form of cosmologically distributed compact objects , since the mean of the density inhomogeneities should be zero , we should compensate the randomly distributed point masses on each plane by a _ negative _ smooth surface mass density . then a term of the form @xmath46 should be added to the right hand side of equation ( [ eq5 ] ) . as we mentioned in   1 , numerical simulations of gravitational microlensing by a three - dimensional random distribution of point masses have revealed differences between two- and three - dimensional microlensing ( refsdal 1970 ; schneider & weiss 1988a , b ; rauch 1991 ) . to find out and understand qualitatively some of these differences , it is useful to study in detail specific cases of simple lens configurations . in this section we study numerically the properties of multiplane gravitational lensing by a few point masses . to reduce the number of parameters , we restrict our study to cases where the distance between the source and the observer is much smaller than the horizon scale , i.e. , @xmath47 . if there are @xmath48 point masses on lens plane @xmath22 ( we shall denote the total number of point masses as @xmath49 ) , the sum of the squared ( lagrangian ) einstein radii is @xmath50 for the numerical computations , it is convenient to express all positions in units of @xmath51 and to define a dimensionless mass parameter @xmath52 so that @xmath53 . then the multiplane lens equation with identical point masses ( eqs.[[eq1 ] ] and [ [ eq5 ] ] ) can be written as @xmath54      first let us consider in detail configurations with two point masses . without any loss of generality , we can choose a coordinate system such that @xmath55 and @xmath56 . since we assume that @xmath47 , the two remaining parameters , @xmath57 and @xmath58 , are related by @xmath59 . the topology of the critical curves ( and the caustics ) as a function of the parameters @xmath60 has been classified by es ( see also schneider & weiss 1986 for the @xmath61 case ) . in general , for given @xmath57 and @xmath58 , there are five transitions in the topology of the critical curves as @xmath62 increases from @xmath63 ( see fig .  6 of es ) . however , in the limit @xmath64 ( i.e. , in the limit of two point masses on the same plane ) , three of the transitions merge at @xmath65 and the number of transitions is reduced to two . for comparison , we show in figures 27 two sets of models with @xmath61 and @xmath66 . ( we have also computed models with @xmath67 , but the results are qualitatively similar to those with @xmath66 and are not shown here . ) the configurations shown in figures 27 were chosen to sample regions of the parameter space with different critical curve topologies . the parameters @xmath68 and the critical curve topology types ( in the notation of es ) of these configurations are listed in table 1 . ( note that the topology type @xmath69 is not represented ; with @xmath59 , type @xmath69 only occurs at very small @xmath62 . )    in figures 2 and 5 we show the critical curves of the configurations with @xmath61 and @xmath70 respectively . the corresponding caustics and iso - magnification contours ( on the observer plane ) are shown in the left and right panels of figures 3 and 6 . in figures 4 and 7 we plot the `` normalized '' differential cross - section @xmath71 , where @xmath0 is the differential cross - section for magnification @xmath72 and @xmath73 is the differential cross - section of a single isolated point mass lens with einstein radius @xmath51 as defined in equation ( [ eq7 ] ) . as we saw in paper i ( see also   4 below ) , the function @xmath74 is a useful measure of the deviations of the cross - section of a more complex lens configuration from that of the point mass lens . the cross - sections and the iso - magnification contours shown in these ( and all subsequent ) figures were obtained using the ray - shooting method ( see , e.g. , kayser , refsdal , & stabell 1986 ; schneider & weiss 1986 , 1988b ) . since the magnification factor is calculated in this method by collecting rays into a grid of square pixels , it is the magnification for an observing device ( or equivalently a source ) of the size of one pixel . to assess the effects of finite pixel size , we performed two sets of calculations , with the linear pixel sizes in one set a factor @xmath75@xmath76 smaller ( see table 1 ) . the pixel sizes were chosen so that all ( except the very small ) caustics are resolved . we show in figures 3 and 6 the contours from the calculations with the smaller pixels and in figures 4 and 7 the function @xmath74 from both sets of calculations . as expected , the finite pixel size introduces an upper limit to the magnification factor and produces extra `` bumps '' in @xmath74 ( rauch  1992 ) , but it does not have significant effects on the position or the strength of the real caustic - induced features in @xmath74 . the @xmath61 ( and @xmath77 ) models have already been studied in detail by schneider & weiss ( 1986 ) ; hence , we only give a brief description of these models here . for @xmath78 , the region near each of the point masses is perturbed by the weak shear from the other point mass . so we have two critical lines that are slightly flattened ellipses and two caustics that have the familiar `` astroid '' shape ( see   3 and fig .  2 of paper i ) . as @xmath79 , the critical curves and the caustics become asymmetric ( figs . 2_a _ and 3_a _ ) . the corresponding differential cross - section ( fig . 4_a _ ) shows caustic - induced features that are similar to those found in the case of a point mass perturbed by shear ( see fig .  3 of paper i ) . there is , in addition , a bump ( at @xmath80 ) which is associated with the last iso - magnification contour that encloses both of the caustics . as @xmath62 decreases below @xmath76 , the number of critical curves ( and caustics ) changes from two to one to finally three , and the topology of the critical curves ( and the caustics ) changes from @xmath81 to @xmath82 to finally @xmath83 ( see table 1 ) . in all cases , there are caustic - induced features in the differential cross - section at moderate to high @xmath72 and the cross - section at small @xmath72 is reduced . each of these features is associated with either a local minimum or saddle point in the magnification pattern or an iso - magnification contour that touches the caustics at one or more points . in the latter case , the contours of slightly larger @xmath72 are broken into disconnected shorter segments by the caustics and the loss of the area inside the caustics results in a drop in @xmath74 . ( an example that has been discussed in detail in   3 of paper i is the feature associated with the contour osculating the astroid - shaped caustic of the point mass plus weak shear lens . )    when the two point masses are not on the same lens plane , there are new features and transitions in the critical curves and the caustics ( see figs .  5 and 6 ) . at large lagrangian separation @xmath62 ( fig . 5_a _ ) , besides the two critical curves that are similar to the ones seen in the @xmath61 case ( see   4 for an explanation ) , there is an extra critical curve inside the critical curve that encloses point mass @xmath84 ( i.e. , the point mass closer to the source ) . this critical curve is offset from point mass @xmath84 in the direction opposite point mass @xmath76 and both its size and its distance from point mass @xmath84 increase with decreasing @xmath62 . the corresponding caustic is located at a large distance along the positive @xmath85-axis ( fig . this caustic has the ( asymmetric ) astroid shape of the two caustics that are known from the @xmath61 case , and it increases in size and moves toward the other two caustics as @xmath62 decreases . the critical curve ( and caustic ) configuration at large @xmath62 that we have just described is denoted as topology type @xmath72 by es . except for the presence of the new critical curve and caustic , the topological transitions shown in figures 5_a__c _ and 6_a__c _ are similar to those shown in figures 2 and 3 . the remaining transitions do not exist in the single lens plane case . however , as es pointed out , except for the elliptic umbilic transition between @xmath86 and @xmath87 , all transitions are the beak - to - beak type . we can explain the new critical line ( and caustic ) and its qualitative behaviors if we focus on the configurations of topology types @xmath72@xmath86 . for these configurations , it is clear that the new critical line is composed of the set of rays that are strongly deflected by point mass @xmath84 ( on lens plane @xmath84 ) to the vicinity of point mass @xmath76 ( on lens plane @xmath76 ) and are then infinitely magnified by point mass @xmath76 . obviously this process can only occur when the point masses are not on the same lens plane . the new critical line is closer to point mass @xmath84 for larger @xmath62 because a light ray must be more strongly deflected by point mass @xmath84 to reach the vicinity of point mass @xmath76 .    in the magnification patterns ( fig . 6 ) , there are local minima and saddle points that are associated with the new caustic ; there are also iso - magnification contours that touch the new caustic at two points ( e.g. , the second outermost contour in fig . these new features in the magnification patterns are expected to produce corresponding features in the cross - sections . however , as we can see in figure 7 , the `` normalized '' differential cross - sections @xmath74 are actually quite similar to those shown in figure 4 . features associated with the new caustic are not noticeable until @xmath88 , and even in cases where the new caustic ( which is transformed at small @xmath62 ) encloses a large area of the observer plane , the new features are typically weak and at @xmath89 . ( they are stronger and at higher @xmath72 for the @xmath67 models . ) if we examine the magnification patterns ( and fig . 8) more carefully , we see that the new caustic is usually located in regions where the background magnification is small . in addition , the magnification throughout most of the region inside the new caustic and , in particular , the minimum magnification inside the new caustic are only slightly higher than the low background magnification . hence the new features in the magnification patterns and the corresponding features in @xmath74 have small @xmath72 values . next , let us consider specifically the feature associated with the iso - magnification contour that touches the new caustic at two points ( e.g. , the second outermost contour in fig . this contour is long and , except for the distortion near the new caustic , nearly circular . contours of slightly higher magnification are broken by the new caustic into a short arc around one of the cusps and a long arc that encloses the other caustic(s ) . this results in a decrease in the cross - section @xmath0 because an area that would otherwise be enclosed by two of these contours ( of magnification @xmath72 and @xmath90 ) is lost to the interior of the caustic . although the area lost to the caustic can be large , it is typically much smaller than the area enclosed by the long ( and the short ) arcs . consequently , the decrease in @xmath91 , which is approximately the fractional decrease in @xmath0 , is small . similarly , the increase in @xmath74 at @xmath92 , where @xmath93 is the minimum magnification inside the new caustic , is small because @xmath93 is typically small and the area enclosed by the contours of magnification @xmath93 and @xmath94 outside the new caustic ( mainly two long arcs ) is much larger than the area enclosed by the contour @xmath94 inside the new caustic . the relatively weak influence of the new caustic on the magnification pattern is demonstrated more clearly in the sample of magnification profiles plotted in figure 8 . although the magnification must be infinite on the caustic , only a very small region near the caustic has @xmath95 . in fact , the numerically obtained @xmath72 , which is finite at a caustic because of the finite pixel size of our calculations , shows a maximum at the new caustic that is quite small . this behavior of the magnification pattern near the new caustic matches the description of the `` thin caustics '' found in the microlensing simulations of schneider & weiss ( 1988b ) . we have therefore found in the simple two - point - mass case a clear demonstration of the occurrence of thin caustics . as we mentioned earlier , the new caustic consists of light rays that are strongly deflected by the first point mass to the vicinity of the second point mass and are then infinitely magnified by the second point mass . in the simulations of schneider & weiss , which have a large number of point masses distributed on several lens planes , a light ray can be strongly deflected on more than one lens plane . it is likely that most of the caustics that are strongly deflected on one or more lens planes ( i.e. , the secondary caustics defined in   3.2 below ) are thin caustics . this identification for the thin caustics is consistent with the conjecture by schneider & weiss that the thin caustics are caused by steep deflection gradients because the regions ( in lagrangian space ) with strong deflections are also regions with steep deflection gradients . now let us consider briefly configurations with three and four point masses . two configurations with @xmath96 point masses and one with @xmath97 are shown in figures 911 . these configurations have one point mass on each lens plane ( i.e. , @xmath98 ) and their parameters are listed in table 2 . the configuration _ a _ with @xmath99 was chosen because it has the critical curve topology of point masses widely separated in lagrangian space while the smaller critical curves are clearly resolved . the other configurations are `` typical '' configurations resulting from randomly distributing the point masses within a cone of constant lagrangian radius . we have ( arbitrarily ) chosen a cone radius such that a light ray passing within the cone would be infinitely magnified if the mass of the lenses were uniformly distributed within the cone .    a striking feature of figure 9 ( and fig . 5 ) is the rapid increase in the number of critical curves with @xmath1 . for a configuration with @xmath1 point masses well separated in lagrangian space and with each point mass on its own lens plane ( i.e. , @xmath98 ) , we can calculate the number of critical curves by extending the discussion given in   3.1 for the origin of the new critical curve . light rays from the source can reach the vicinity of the point mass on the @xmath22th lens plane after being strongly deflected by any combination ( or none ) of the previous @xmath100 point masses . for each combination , there is a set of rays that are infinitely magnified by the @xmath22th point mass ( and the weak deflections on the remaining planes ) . therefore , there are @xmath101 critical curves associated with the @xmath22th point mass and the total number of critical curves is @xmath102 . the critical curves shown in figures 5_a _ and 9_a _ are consistent with these numbers . ( the total number of critical curves is smaller for configurations with @xmath103 and approaches @xmath1 in the limit @xmath104 , i.e. , in the single lens plane case . ) hereafter , we shall refer to the caustics that are not strongly deflected as the _ primary _ caustics and those that are strongly deflected on at least one lens plane as the _ secondary _ caustics . although the number of critical curves increases exponentially with @xmath1 , we can see in figure 9 that the size of a critical curve decreases rapidly as the number of strong deflections associated with the critical curve ( or equivalently the number of critical curves within which it is nested ) increases . on the other hand , as in the two - point - mass case , a caustic formed after a larger number of strong deflections is not necessarily smaller . for example , in figure 10_c _ , the caustic located near @xmath105 [ which corresponds to the critical curve located near @xmath106 and the caustic with six cusps [ which corresponds to the largest critical curve ] are comparable in size .    in the two - point - mass case , due to the absence of overlapping or self - intersecting caustics , a complete classification of the critical curve topology also provides a complete classification of the caustic topology ( es ) . this is not true when @xmath107 . in particular , there are overlaps of caustics that belong to different critical curves . examples of such overlapping caustics are the caustics located near @xmath108 in figure 10_a _ and those in the lower left corner of figure 10_c_. ( there are no self - intersecting caustics in the configurations shown in fig .  10 , but they may be present in other configurations . )    finally , we discuss the normalized differential cross - sections plotted in figure 11 . for the `` typical '' configurations _ b _ and _ c _ , the qualitative behaviors of @xmath74 agree with what one might expect from the two - point - mass configurations studied in   3.1 . the cross - section at small @xmath72 is smaller than that for a single point mass . there are prominent features at moderate to high @xmath72 that are associated with the primary caustics . features associated with the secondary caustics are typically weak , but some of them are noticeable at @xmath89 . for the configuration _ a _ , the features associated with the primary caustics , i.e. , those at @xmath109 , are similar to the features in figures 4_a _ and 7_a_. this is expected since configuration _ a _ has the critical curve topology of widely separated point masses . however , the features associated with the secondary caustics are at unusually high @xmath72 : the `` plateau '' between @xmath110 and @xmath111 consists of a series of such features . this is due to the fact that the parameters of configuration _ a _ were specially chosen so that the smaller critical curves are clearly resolved . for these parameters , the caustics are unusually close to one another ( see fig . 10_a _ and notice the change in scale between figs . 10_a _ and 10_b _ ) . as a result , the secondary caustics are located in regions of relatively high background magnification and the associated features in @xmath74 are at unusually high @xmath72 . the study of simple lens configurations in the previous section shows that microlensing by a three - dimensional distribution of point masses is , in general , much more complicated than microlensing by lenses confined to a single lens plane . but there are also indications that , in the low optical depth limit , some aspects of the three - dimensional problem can be reduced to the planar one . for example , the multiplane configurations with point masses widely separated in lagrangian space show primary caustics and corresponding features in @xmath74 that are similar to those found in the single plane configurations . in   4.1 we confirm this analytically for the limiting case of a point mass lens perturbed by the weak shear from the other lenses . another useful result from the study in 3 is that the significant features in @xmath74 at moderate to high @xmath72 are associated with the primary caustics . therefore , we can neglect the features associated with the secondary caustics when we analyze the macroimage magnification distribution , @xmath4 , at high @xmath72 . in   4.2 we derive the feature induced by the primary caustics at the high magnification end of @xmath4 for a three - dimensional low optical depth lens distribution . the resulting semi - analytic @xmath4 is compared to previous numerical results in   4.3 , and the differences between the two- and three - dimensional results are discussed in   4.4 . in the low optical depth limit , lensing ( other than low magnification events ) by a three - dimensional distribution of point masses is , as in the planar case , typically dominated by a single point mass . without loss of generality , we shall designate one of the point masses residing on the @xmath112th lens plane as the dominant one and locate this point mass at the origin of the lens screen . on all of the other screens , the light ray is only weakly perturbed , and the perturbations are expanded to the first order in @xmath113 . before the light ray passes the @xmath112th screen with the dominant lens ( i.e. , for @xmath114 ) , the lens mapping ( eqs.[[eq1 ] ] and [ [ eq5 ] ] ) is simplified to @xmath115 where the constant deflection    @xmath116    and the shear matrix @xmath117    ( similar to @xmath118 and @xmath119 defined in eq.[13 ] of paper i ) represent the perturbative influence of the lenses on the @xmath22th plane . equation ( [ eq11 ] ) can also be written as @xmath120 where @xmath121 and @xmath122 is the identity matrix . once the light ray has traversed the @xmath112th lens screen , the location at which the light ray intersects subsequent ( i.e. , @xmath123 ) screens is given by @xmath124 where @xmath125 denotes the influence of the dominant lens and @xmath126 note that , in the absence of the dominant lens , @xmath127 is the path followed by a light ray with @xmath128 .    on the observer plane [ which is the @xmath129th screen , but we also denote it with the subscript `` o '' ] , the eulerian position is given by @xmath130 if we use the fact that @xmath131 and transform the eulerian variable according to @xmath132 , the above equation can be cast into the form    @xmath133    where @xmath134 , @xmath135    although there is only shear on each lens screen , the matrix @xmath136 is in general a combination of shear and convergence ( i.e. , it is symmetric but not traceless ) . thus equation ( 19 ) is formally equivalent to the single plane lens equation for the point mass lens plus shear and smooth surface density ( compare with eq.[15 ] of paper i for the point mass plus shear lens ) . this is a specific example of the `` non - linear telescope '' ( or generalized quadrupole lens ) discussed by kovner ( 1987 ; see also schneider  1992 ) . we are interested in the case where the shear perturbations on all the lens screens are weak . to first order in @xmath137 , the expressions for @xmath138 , @xmath139 , and @xmath136 are    @xmath140    and @xmath141    respectively . hence @xmath136 is reduced to a shear matrix : @xmath142 , where @xmath143 is the magnitude of the `` effective '' shear and @xmath144 then equation ( 19 ) reduces to the single plane lens equation for the point mass plus weak shear lens ( eq.[15 ] of paper i ) .    in    3.12 of paper i , we studied in detail the properties of single plane lensing by the point mass plus weak shear lens . the results are applicable to the mapping from @xmath145 to @xmath146 ( eq.[19 ] ) . we can then infer the properties of the lens mapping from @xmath15 to @xmath16 ( eq.[[eq18 ] ] ) by taking into account the effects of the transformations @xmath147 and @xmath148 . first , it is easy to see that , to first order in @xmath137 , both transformations are a combination of translation and shear distortion and are one - to - one . second , since the deformation tensor @xmath149 there is a simple correspondence between the caustics ( and the critical curves ) of the mappings ( [ eq18 ] ) and ( 19 ) . in addition , because the determinants @xmath150 and @xmath151 are @xmath152 , the magnification @xmath153 at a point @xmath16 is , to first order in @xmath143 , identical to the `` magnification '' @xmath154 at the corresponding point @xmath155 . similarly , the correction factors for the differential cross - section are @xmath152 . all of the corrections that we have just considered are of the same order as the terms already ignored in the weak shear analysis in paper i. therefore , the effects of the transformations are negligible , and the properties derived in    3.1 - 2 of paper i are also applicable to the lens mapping ( [ eq18 ] ) . briefly , for a point mass lens perturbed by weak shear in three dimensions , the critical line is a slightly flattened ellipse ( of size @xmath156 ) ; the caustic has the shape of an astroid , with the four cusp catastrophes ( at a distance @xmath157 from the center of the astroid ) connected to each other by four fold catastrophes ( see fig .  2 of paper i ) ; the differential cross - section has the scaling property ( see eq.[23 ] of paper i ) @xmath158 ) = { 2 \pi x_{e , l}^2 \over ( a^2 -1)^{3/2 } } \ ,                            \varphi(a s'[\chi_{lo } ] ) , \label{eq23}\ ] ] where the normalized differential cross - section @xmath159 depends on @xmath72 and @xmath143 only through the combination @xmath160 ; and @xmath159 , which is shown in figure 3 of paper i , shows strong caustic - induced features at the critical values @xmath161 and @xmath162 .    as we mentioned in   2 , when we consider the situation where a significant fraction of the dark matter in the universe is in the form of cosmologically distributed compact objects , we should compensate the randomly distributed point masses on each lens plane by a negative smooth surface mass density . the derivation in this section can be generalized to this case if we replace @xmath137 in all the equations ( except eq.[[eq12b ] ] ) by @xmath163 . the results remain valid , but the magnification @xmath72 should be interpreted as the magnification with respect to an `` empty beam '' ( i.e. , a light beam that is not affected by the point masses ) . as we discussed earlier , in the case of low optical depth microlensing by a three - dimensional distribution of point masses , features associated with the secondary caustics are negligible at the high magnification end of the macroimage magnification distribution @xmath4 . therefore , as in the two - dimensional case ( see   3.3 of paper i ) , @xmath4 at high @xmath72 can be approximated as a superposition of the cross - sections of the individual point mass lenses , with the shear perturbation due to the other point masses varying from lens to lens . let us assume that the point masses are distributed randomly on @xmath17 lens planes , each with projected ( onto the observer plane ) surface number density @xmath164 . then @xmath4 can be written as @xmath165 where @xmath166 is the cross - section given in equation ( [ eq23 ] ) , @xmath167 is the probability distribution of the effective shear @xmath143 for a dominant lens at @xmath168 , @xmath169 is the probability that the dominant lens is located on the @xmath112th screen , and @xmath170 is the total projected surface number density . since @xmath169 is simply @xmath171 we have @xmath172 where @xmath173 the function @xmath174 has the same structure as @xmath4 of the two - dimensional case ( eq.[29 ] of paper i ) , and it is the contribution to the macroimage magnification distribution by the lenses on the @xmath112th lens screen as the dominant lenses .    the effective shear @xmath143 that enters into the cross - section @xmath166 is a weighted superposition of the shear induced by all except the dominant lens ( eq.[[eq20b ] ] ) . in the appendix we show that @xmath175^{3/2 } } , \label{eq28}\ ] ] where @xmath176 is the `` effective '' optical depth at @xmath168 . [ note that the effective optical depth @xmath177 is smaller than the ( usual ) optical depth @xmath178 because the extra factors @xmath179 ( for @xmath180 ) and @xmath181 ( for @xmath182 ) are less than unity . ] substituting equations ( [ eq23 ] ) and ( [ eq28 ] ) into equation ( [ eq27 ] ) , we find that @xmath174 is similar to @xmath4 for a single lens plane ( compare with eq.[31 ] of paper i ) : @xmath183,\ ] ] where @xmath184 is the function introduced in   3.3 of paper i to describe the caustic - induced feature in @xmath4 in the single lens plane case ( see fig .  4 of paper i ; also fig . 12 of this paper ) . thus @xmath174 exhibits the same caustic - induced bump ( with a maximum enhancement of @xmath185 at @xmath186 ) and the same scaling property ( with @xmath187 being a function of the combination @xmath188 ) as @xmath4 of the single lens plane case . there is , however , a significant difference between the two- and three - dimensional cases . to obtain the macroimage magnification distribution in the three - dimensional case , we must sum @xmath174 over all the lens planes ( eq.[[eq26 ] ] ) : @xmath189       = { 2 \tau \over ( a^2 - 1)^{3/2 } } f_n ( a ; \tau ) , \label{eq31}\ ] ] where @xmath190 , \label{eq32}\ ] ] and @xmath191 is the ( usual ) optical depth . although we denote the caustic - induced feature in the @xmath17-plane case as @xmath192 for simplicity , it should be clear from equation ( [ eq32 ] ) that @xmath192 depends on the location and the surface density of the lens planes in a non - trivial manner and does not have any simple scaling properties ( when @xmath193 ) . note also that the differences between @xmath194 and @xmath192 are due to the differences between the optical depths @xmath11 and @xmath177 .    to proceed further , we need to assume a model for the distribution of the point mass lenses . as in the numerical simulations reported by several authors ( refsdal 1970 ; schneider & weiss 1988a , b ; rauch 1991 ) , we consider a lens distribution with _ constant _ comoving mass density @xmath195 . we divide the volume between the source and the observer into @xmath17 shells of equal comoving thickness ( @xmath196 ) and place the point masses in each shell on a lens plane at the middle of the shell . in figure 12 we plot the numerically evaluated @xmath192 for @xmath104 , @xmath76 , @xmath197 , @xmath198 , and @xmath199 , and for source redshift @xmath200 and @xmath201 . it is clear that @xmath202 has converged by @xmath203 . this agrees with the conclusion from a study of lensing by smoothly variable mass distribution that a fully three - dimensional mass distribution is well approximated by as few as eight or @xmath199 screens ( lee & paczyski 1990 ) . it is also clear that @xmath204(@xmath205 ) is very insensitive to the source redshift , at least for @xmath206 .    in the limit @xmath207 , the summations in the expressions for @xmath11 , @xmath177 , and @xmath192 are replaced by integrals :    @xmath208 ; \\ f_\infty ( a ; \tau ) & = & { 4 \pi g \rho_l a_0 ^ 3 \over \tau }                     \int_0^{\chi_{so } } d\chi                     { \chi \left(\chi_{so}-\chi\right ) \over a(\chi ) \chi_{so } }                     f_1 \left[a \tau { \tau ' \over \tau}\right ] .\end{aligned}\ ] ]    equations ( 33a  b ) can be evaluated analytically , and they are particularly simple in the limit @xmath200 ( i.e. , in the limit @xmath209 ) :    @xmath210 .\end{aligned}\ ] ]    thus the contribution to @xmath204 is dominated by the lens planes at @xmath211 with @xmath212 . this is consistent with the numerical results shown in figure 12 . the most significant change from @xmath187 to @xmath204 is the shift in the location of the bump from @xmath213 to @xmath214 . otherwise , the peak value ( @xmath215 enhancement ) and the width of @xmath204 are only slightly smaller and wider . several authors ( refsdal 1970 ; schneider & weiss 1988a , b ; rauch 1991 ) have used numerical simulations to calculate the macroimage magnification distribution , @xmath4 , produced by a three - dimensional distribution of point masses . in the most recent study by rauch , distributions with high resolution in @xmath72 were obtained from monte carlo simulations for three cases with @xmath216 , @xmath217 , and @xmath218 . the `` equivalent @xmath219 value '' is related to the average magnification ( with respect to an empty beam ) on the observer plane , @xmath220 , by @xmath221 , i.e. , @xmath219 is the optical depth @xmath11 of a _ two - dimensional _ lens distribution with the same @xmath220 . for a three - dimensional lens distribution , @xmath219 and @xmath11 are not identical in general , but @xmath222 in the limit @xmath223 . we wish to compare the semi - analytic @xmath4 derived in   4.2 to the distributions obtained by rauch . ( we shall focus on his results for an einstein - de sitter cosmological background , but rauch also found that the distributions for the same @xmath219 but different background cosmology are nearly identical . ) since our derivation assumes low optical depth , the best case for comparison should be the @xmath216 case . unfortunately , even with @xmath224 simulations , the monte carlo simulations only provide useful information about @xmath4 for @xmath225 ( see fig .  2 of rauch 1991 ) . this is not sufficient for detecting the caustic - induced feature because significant deviation from @xmath226 is not expected for @xmath227 ( @xmath228 for @xmath229 ) ( see @xmath204 in fig . we shall not show the comparison for this case and simply note that the numerically obtained @xmath4 ( for @xmath225 ) agrees with the low optical depth semi - analytic result , with or without the caustic - induced feature .    in figure 13 we show the distributions obtained by rauch for @xmath230 and @xmath218 ( histograms ) . the data are those shown in figures 3 and 4 of rauch ( 1991 ) , but they are plotted in the form @xmath231 and we have increased the bin sizes at large @xmath72 to reduce statistical fluctuations . as rauch  ( 1992 ) pointed out , the distributions in the three - dimensional case do not show the prominent bump at @xmath232 that was found in the planar case . however , the distributions are not featureless . this can be seen most clearly in the @xmath233 case ( note , e.g. , the inflection point at @xmath234 ) .    in a plot of @xmath231 , the low optical depth distribution without the caustic - induced feature is a horizontal line of amplitude @xmath235 , while the semi - analytic distribution with the caustic - induced feature is @xmath236 ( eq.[[eq31 ] ] ; recall that @xmath204 is very insensitive to the source redshift ) . since @xmath230 and @xmath218 are not much less than unity , two corrections to the low optical depth semi - analytic results are necessary when we compare them to the numerical ones . first , it is no longer accurate to assume that @xmath237 . for these particular models , @xmath230 and @xmath218 correspond to @xmath238 and @xmath239 , respectively . second , the normalization @xmath240 can be significantly different from the low optical depth result @xmath241 . in 5.1 of paper i , we discussed the renormalization of @xmath4 due to the net convergence of light rays and multiple streaming for the single lens plane case . from an analytic result for @xmath4 in the high magnification limit , which is exact even for finite optical depth @xmath11 ( see also schneider 1987 ) , we adopted the normalization @xmath242 ( for @xmath223 , @xmath243 and @xmath244 , and the correction to the normalization @xmath241 is small . ) the analysis that led to equation ( [ eq35 ] ) can not be easily generalized to the three - dimensional case ( it would require the statistical properties of the matrix @xmath245 which depends on the shear matrices @xmath137 _ non - linearly _ ; see eq.[14 ] ) , and an equivalent expression for @xmath240 for a three - dimensional lens distribution is not known . nevertheless , in the range of @xmath11 of interest here ( i.e. , for @xmath246 ) , it is likely that equation ( [ eq35 ] ) is a good approximation for @xmath240 in the three - dimensional case . in equation ( [ eq35 ] ) , the dominant correction to the normalization @xmath241 is the factor @xmath220 . ( the correction due to the factor @xmath247 is second order in @xmath11 and less than @xmath248 for @xmath246 . ) the factor @xmath220 arises from converting a probability distribution in lagrangian space to a distribution in eulerian space ( see   5.1 of paper i ) . the conversion ( and hence the factor @xmath220 ) should remain in any generalization of the analysis to the three - dimensional case . furthermore , as in the planar case ( see   5.1 of paper i ) , we can derive the factor @xmath220 using a different argument : in   4.2 we assumed that the surface density of the astroid - shaped caustics on the observer plane is the same as the projected surface density of point masses @xmath249 ( see eq.[[eq24 ] ] ) ; there is , however , a net convergence of the light rays by the overall lens distribution ; this increases the density of astroids by a factor @xmath220 , and introduces an additional factor @xmath220 in the normalization of @xmath4 in equation ( [ eq31 ] ) . therefore , we shall adopt equation ( [ eq35 ] ) for the normalization @xmath240 in the three - dimensional case .    in figure 13 we plot @xmath240 ( dashed lines ) and @xmath236 ( solid lines ) for @xmath230 and @xmath218 . they should be compared to the numerical results at @xmath250 ( see   4.5 for a brief discussion of the deviations at small @xmath72 ) . as we can see , the adopted normalization @xmath240 is in good agreement with the amplitudes of the numerical results . in addition , the numerical results show deviations from the horizontal dashed lines that are similar in shape to the semi - analytic caustic - induced feature @xmath204 ( note , e.g. , that the inflection point in the numerical result for @xmath233 coincides with the location @xmath251 where @xmath204 starts to deviate significantly from unity ) . in the @xmath230 case , there is little , if any , difference between the numerical and semi - analytic results ; the only possibly statistically significant deviation is that the numerical distribution may be higher near @xmath252 . in the @xmath233 case , the difference between the numerical and semi - analytic results is more pronounced , with the numerical distribution slightly higher than the semi - analytic distribution over the range @xmath253 . note , however , that the number of events in the histograms at @xmath254 is small ( see the fluctuations amongst the same bins in the @xmath230 case ) , and the true distribution may not be as flat as the histograms shown .      the comparison in 4.3 shows that , in the case of a three - dimensional lens distribution , the function @xmath204 , which was derived in the low optical depth limit , provides a reasonably good fit to the caustic - induced feature in @xmath4 for @xmath11 as large as @xmath239 . this is very different from the two - dimensional case studied in paper i. for a two - dimensional lens distribution , the caustic - induced feature in the numerically obtained @xmath4 is more prominent than the low optical depth semi - analytic feature @xmath187 for @xmath11 as small as @xmath217 . this was shown in figure 6 of paper i where we compared the numerical and semi - analytic distributions for @xmath255 and @xmath218 in a plot of @xmath256 . the numerical distributions show a clear minimum at @xmath257 and are significantly higher than the semi - analytic results in the range @xmath258 . for both the two- and three - dimensional lens distributions , the low optical depth analysis assumes that the point mass lenses are well separated in lagrangian space . then the deflection near each point mass is due to that point mass and the shear perturbation from the other lenses , and an astroid - shaped ( primary ) caustic is associated with each point mass . the low optical depth analysis is not valid if the lens density is high enough that a significant fraction of the lenses have close neighbor(s ) , and their caustics are complex structures produced by the collective effect of two ( or more ) point masses . in 5.3 of paper i , we estimated for the two - dimensional case the fraction of point masses whose caustics are not isolated astroids , @xmath259 , by considering a point mass and its nearest neighbor as a two - point - mass lens : @xmath260 . for @xmath6 , @xmath261 . this explains the relatively strong contribution to @xmath4 by the collective effect of two ( or more ) point masses ( which results in the enhanced caustic - induced feature at moderate @xmath72 ) for @xmath11 as small as @xmath217 . to understand the difference between the two- and three - dimensional cases , we need to estimate @xmath259 for the three - dimensional case . we now extend the method used in paper i to estimate @xmath259 to the three - dimensional case . we begin by considering lensing configurations with two point masses on different lens planes . since our results in   4.2 and the numerical results of rauch ( 1991 ) show that @xmath4 is not sensitive to either the source redshift or the background cosmology , we shall simplify the analysis by assuming that @xmath209 . then the two point mass configurations of interest are identical to those studied in   3.1 and are specified by two parameters : the dimensionless distance @xmath24 between the lens planes and the dimensionless lagrangian separation @xmath62 between the lenses [ the actual lagrangian separation is @xmath262 ; see eq.([eq7 ] ) ] . note that we have changed the notation slightly and the point masses are on lens planes @xmath22 and @xmath19 instead of lens planes @xmath84 and @xmath76 . as we saw in   3.1 , the topology of the caustics ( and the critical curves ) changes with decreasing @xmath62 from type @xmath72 to @xmath69 . the only topology with two astroid - shaped primary caustics is type @xmath72 ; the other topology types are produced by the strong effect of both point masses ( see fig .  6 ) . in figure 14 we plot the separation @xmath263 at which the topology changes from type @xmath72 to type @xmath264 as a function of @xmath24 . ( the data were obtained using the numerical code that generated the caustics and critical curves shown in figs . the separation @xmath263 decreases with increasing @xmath24 . in the limit @xmath265 , the two point masses are on the same plane and we have @xmath266 ( schneider & weiss 1986 ) . in the limit @xmath267 ( i.e. , in the limit @xmath268 or @xmath269 ; see eq.[[eq2 ] ] ) , @xmath270 ; this is due to the focusing of the light rays by the point mass on lens plane @xmath22 and the fact that @xmath271 . let us consider first a simple extension of the method used in paper i to estimate @xmath259 . it is convenient to consider the projected distribution ( i.e. , the distribution in lagrangian space ) of the point mass lenses . if we select randomly one of the point masses , the probability that the point mass is on lens plane @xmath22 is @xmath272 , where @xmath170 is the total projected surface number density of point masses . this point mass has an absolute nearest neighbor in lagrangian space . since the point masses are distributed randomly , the probability that the absolute nearest neighbor is on lens plane @xmath19 and at a lagrangian distance less than @xmath273 is @xmath274 $ ] . if we ignore the deflection due the other lenses and consider the point mass and its nearest neighbor as a two - point - mass lens , the discussion in the previous paragraph tells us that the primary caustic associated with the point mass is not an isolated astroid if @xmath275 . therefore , an estimate of the fraction of point masses whose caustics are not isolated astroids is @xmath276 \right\ } . \label{eq36}\ ] ]    in the three - dimensional case , since the ( lagrangian ) einstein radius @xmath156 increases from @xmath63 to @xmath277 as @xmath168 increases ( see eq.[[eq6 ] ] ) , it is possible , e.g. , for the primary caustic ( and critical curve ) of a point mass to merge with that of its second nearest neighbor ( from plane @xmath36 ) , but not with that of its absolute nearest neighbor ( from plane @xmath19 ) , if @xmath278 . thus the function @xmath279 , which takes into account the absolute nearest neighbor only , probably underestimates the fraction of point masses whose caustics are not isolated astroids . we can derive a better estimate by taking into account the nearest neighbors from all of the lens planes . for a randomly selected point mass ( on lens plane @xmath22 with probability @xmath272 ) , the probability that the nearest neighbor from lens plane @xmath19 is at lagrangian distance greater than @xmath273 is @xmath280 . if we ignore the deflection due the other lenses and consider the point mass and its nearest neighbor from lens plane @xmath19 as a two - point - mass lens , the two - point - mass lens produces two astroid - shaped primary caustics if @xmath281 . thus the probability that the primary caustic associated with the point mass is an isolated astroid is @xmath282 $ ] . therefore , a better estimate of the fraction of point masses whose caustics are not isolated astroids is @xmath283 \right\ } . \label{eq37}\ ] ]    as with @xmath192 , although @xmath284 and @xmath285 are denoted as functions of @xmath11 , they do depend on the location and the surface density of the lens planes . for @xmath104 , i.e. , for a single lens plane , both @xmath286 and @xmath279 reduce correctly to @xmath287 . in the limit @xmath223 , @xmath288 in figure 15 we show @xmath279 for @xmath104 , @xmath76 , @xmath197 , @xmath198 , and @xmath277 ( dashed lines ) and @xmath289 ( solid line ) , using the constant comoving density model adopted in   4.2 . both @xmath284 and @xmath285 have converged by @xmath290 . the difference between @xmath291 and @xmath292 is small for @xmath293 , but it increases to @xmath294 at @xmath295 .    it is clear from figure 15 that @xmath289 is significantly smaller than @xmath296 ( the highest dashed curve in fig . 15 ) . in the limit @xmath223 , @xmath297 while @xmath298 . for @xmath6 , @xmath299 and @xmath261 ; even for @xmath300 , @xmath301 only . thus , in the three - dimensional case , the fraction of point masses whose ( primary ) caustics are not isolated astroids and the contribution to @xmath4 by configurations of two ( or more ) point masses are much smaller . this explains why the low optical depth semi - analytic feature @xmath204 provides a reasonably good fit to the caustic - induced feature in @xmath4 for @xmath11 as large as @xmath239 . at sufficiently large @xmath11 ( @xmath302 ) , @xmath303 and @xmath4 should deviate significantly from the semi - analytic result . finally , we discuss briefly the deviations of @xmath4 from the semi - analytic distribution @xmath304 at low magnification ( see fig . 13 ) . as in the planar case , the superposition of the cross - sections of individual lenses that results in the caustic - induced feature @xmath204 is only valid for the magnifications seen by the observers close to one of the astroid - shaped primary caustics . the semi - analytic distribution also requires modification at low magnification as it diverges as @xmath305 at @xmath306 and is not normalizable . in the planar case , an observer not particularly well aligned with any single lens sees a primary image close to the unperturbed position of the source and a large number of faint secondary images ( one close to each lens ) produced by strongly deflected rays . in paper i , we took into account the correlations between the magnifications of the primary and the secondary images and derived the modification of @xmath4 at low magnification . we have not attempted a similar calculation in the three - dimensional case because the problem is significantly more complicated . first , for an observer that is not particularly close to either a primary or a secondary caustic , there are now @xmath307 secondary images ( where @xmath1 is the number of lenses ) because a light ray can reached the observer after being strongly deflected by one or more lenses . although the secondary images that have been strongly deflected by more than one lens are typically much fainter than those that have been strongly deflected by only one lens , they are far more numerous and we may have to account for them . a more serious problem is the presence of weak caustic - induced features at low magnification that are associated with the secondary caustics ( see   3 ) . unlike the features associated with the primary caustics at high magnification , there are no simple semi - analytic descriptions of the features associated with the secondary caustics . in this paper and paper i , we have sought to develop a systematic theory of gravitational microlensing . we focused on microlensing by a planar distribution of point masses in paper i. in this paper we have studied gravitational microlensing by a three - dimensional distribution of point masses , focusing on the nature of the critical curves , the caustics , the illumination pattern , and especially the macroimage magnification distribution @xmath4 . in the process , we have explained many of the differences between two- and three - dimensional microlensing that have been seen in previous numerical simulations .    in   3 we studied the lensing properties of three - dimensional lens distributions by considering in detail multiplane configurations with two , three , and four point masses . the lensing properties of a three - dimensional lens distribution is fundamentally different from that of a planar distribution in that a light ray could have arrived in the vicinity of a lens either having suffered only weak deflections along the way from the source , or after having been strongly deflected one or more times by the intervening lens . the latter situation can not arise in a purely planar distribution of lenses , and it gives rise to secondary caustics and critical lines that have no analogues in the planar case . in addition , in the case of more than two point masses , the caustic topology is no longer simply related to the critical curve topology because caustics corresponding to different critical lines may overlap each other .    for @xmath1 point masses that are widely separated in lagrangian space , there are @xmath3 primary critical curves ( i.e. , critical curves that have not suffered any strong deflections ) and @xmath2 critical curves in total . features in the `` normalized '' differential cross - section , @xmath74 , that are associated with the secondary caustics are relatively weak and at low magnifications . significant features occuring at moderate to high magnifications are invariably associated with the primary caustics .    in 4 we derived a semi - analytic expression for the caustic - induced feature at the high magnification end of @xmath4 for a low optical depth lens distribution . in the low optical depth limit , we showed that the multiplane lens equation near a point mass is formally equivalent to the single plane equation of a point mass perturbed by shear . consequently , the ( primary ) caustic has the familiar astroid shape , and the differential cross - section exhibits the same scaling behavior found in paper i ( i.e. , @xmath308 depends on magnification @xmath72 and the effective shear @xmath309 through the combination @xmath310 ) . by modeling the illumination pattern as a superposition of the patterns due to individual `` point mass plus weak shear '' lenses , we found that the caustic - induced feature in the macroimage magnification distribution @xmath4 can be written as a weighted sum of the semi - analytic feature derived in paper i for a planar lens distribution ( eq.[[eq32 ] ] ) . the resulting semi - analytic caustic - induced feature is similar to the feature in the planar case , but it is shifted to higher magnification .    in the moderate to high @xmath72 regime , our semi - analytic @xmath4 agrees remarkably well with the magnification distribution obtained from numerical simulations by rauch ( 1991 ) , even for optical depth as high as @xmath311 . this is very different from the two - dimensional case studied in paper i. in the two - dimensional case , we have found that the caustic - induced feature in @xmath4 tends to be more pronounced than the low optical depth result , even for optical depth as low as @xmath5 . we argued that this particular difference between two- and three - dimensional microlensing is due to the fact that for a given value of optical depth , the fraction of point masses whose ( primary ) caustics are not simple astroids is significantly higher in the planar case than in the three - dimensional case . finally , we discussed briefly the deviations of the numerical @xmath4 from the semi - analytic distribution at low magnification . we thank k.  rauch for providing us with the results of rauch ( 1991 ) . we also thank s.  mao and k.  rauch for helpful discussions . a.b .  and l.k acknowledge the hospitality shown to them at cita during their visits in 1995 . this work was supported in part by nserc ( canada ) , the ciar cosmology program , cita , the institute for astronomy ( l.k . ) , the dudley observatory ( a.b . ) , and a cita national fellowship ( m.h.l . ) . in this appendix we evaluate @xmath167 , the probability distribution of the effective shear @xmath143 at a given @xmath168 , due to a low optical depth three - dimensional distribution of point masses . it is assumed that the point masses are distributed randomly on @xmath17 lens planes , each with projected ( onto the observer plane ) surface number density @xmath164 . specifically , on each lens plane , @xmath312 point masses are distributed randomly within a circle of lagrangian radius @xmath313 .    at a given @xmath168 , the effective shear ( in matrix form ) is ( eq.[[eq20b ] ] ) @xmath314 where ( eq.[[eq12b ] ] ) @xmath315 it is convenient to represent the two distinct components of the shear matrix @xmath137 as a vector : @xmath316 where @xmath317 . then the effective shear @xmath136 can be written as an appropriately weighted vector superposition : @xmath318 where we have defined @xmath319 to simplify the notations .    since all the @xmath320 are statistically independent and have similar statistical properties , the formal expression for the probability distribution of @xmath321 is @xmath322 where @xmath323 is the two - dimensional delta function . to evaluate equation ( a5 ) , it is convenient to express the delta function and @xmath324 as fourier transforms : @xmath325 @xmath326 where the characteristic function @xmath327 is ( nityananda & ostriker 1984 ) @xmath328 substituting equations ( a6 ) and ( a7 ) into equation ( a5 ) , we can perform the integration with respect to @xmath320 : @xmath329 next , we integrate with respect to @xmath330 , grouping together the @xmath48 integrals associated with the @xmath22th screen : @xmath331 in the limit @xmath332 . hence , @xmath333 where the `` effective '' optical depth @xmath177 is @xmath334 performing the integration with respect to @xmath335 in equation ( a11 ) , we obtain @xmath336 since @xmath337 depends only on the magnitude of the effective shear @xmath143 , it is convenient to consider the probability distribution for the latter : @xmath175^{3/2}}.\ ] ] we have expressed the probability distribution explicitly as a conditional probability that depends on the value of @xmath168 . it should be noted that @xmath167 has the same functional form as the shear distribution due to a planar random distribution of point masses ( eq.[30 ] of paper i ) . cccc 0.0 & 2.4 & @xmath81 & 0.020 , 0.0320.0 & 1.0 & @xmath82 & 0.016 , 0.0240.0 & 0.6 & @xmath83 & 0.016 , 0.0240.3 & 2.0 & @xmath72 & 0.020 , 0.0400.3 & 1.0 & @xmath264 & 0.018 , 0.0360.3 & 0.6 & @xmath86 & 0.016 , 0.0320.3 & 0.5 & @xmath87 & 0.016 , 0.0320.3 & 0.4 & @xmath338 & 0.016 , 0.0320.3 & 0.2 & @xmath338 & 0.020 , 0.040    ccclrc a & 3 & 1 & 0.1 & @xmath339 & 0.010 , 0.024 & & 2 & 0.5 & @xmath340 & & & 3 & 0.9 & @xmath341 & b & 3 & 1 & 0.55032 & @xmath342 & 0.012 , 0.024 & & 2 & 0.79370 & @xmath343 & & & 3 & 0.94104 & @xmath344 & c & 4 & 1 & 0.5 & @xmath345 & 0.012 , 0.024 & & 2 & 0.72112 & @xmath346 & & & 3 & 0.85499 & @xmath347 & & & 4 & 0.95647 & @xmath348 &
in paper i we studied the theory of gravitational microlensing for a planar distribution of point masses . in this second paper , we extend the analysis to a three - dimensional lens distribution . 
 first we study the lensing properties of three - dimensional lens distributions by considering in detail the critical curves , the caustics , the illumination patterns , and the magnification cross - sections @xmath0 of multiplane configurations with two , three , and four point masses . for @xmath1 point masses that are widely separated in lagrangian space , 
 we find that there are @xmath2 critical curves in total , but that only @xmath3 of these produce prominent caustic - induced features at the high magnification end of @xmath0 .    in the case of a low optical depth random distribution of point masses , we show that the multiplane lens equation near a point mass can be reduced to the single plane equation of a point mass perturbed by weak shear . 
 this allows us to calculate the caustic - induced feature in the macroimage magnification distribution @xmath4 as a weighted sum of the semi - analytic feature derived in paper i for a planar lens distribution . 
 the resulting semi - analytic caustic - induced feature is similar to the feature in the planar case , but it does not have any simple scaling properties , and it is shifted to higher magnification . the semi - analytic distribution is compared to the results of previous numerical simulations for optical depth @xmath5 , and they are in better agreement than a similar comparison in the planar case . 
 we explain this by estimating the fraction of caustics of individual lenses that merge with those of their neighbors . for @xmath6 , 
 the fraction is @xmath7 , much less than the @xmath8 for the planar case . 
 = cmmib10 scaled 1200 = cmmib10 = cmbsy10 scaled 1200 = cmbsy10    = = = = = =    = `` 710b = ' ' 710c = `` 710d = ' ' 710e = `` 710f = ' ' 7110 = `` 7111 = ' ' 7112 = `` 7113 = ' ' 7114 = `` 7115 = ' ' 7116 = `` 7117 = ' ' 7118 = `` 7119 = ' ' 711a = `` 711b = ' ' 711c = `` 711d = ' ' 711e = `` 711f = ' ' 7120 = `` 7121 = ' ' 7122 = `` 7123 = ' ' 7124 = `` 7125 = ' ' 7126 = `` 7127 = ' ' 7272 = " 7201    # 1#1    # 1