Engineering Fracture Me~~~n~esVol. 43, No. 2, pp. 185-194, 1992 Printed in Great Britain.
TECHNIQUES
OF OPTICAL
0013-7944192 $5.00 + 0.00
Pergamon Press Ltd.
CAUSTICS
PHOTOGRAPHY
D. A. MEYN Code 6327, Naval Research Laboratory, Washington, DC 20375-5000, U.S.A. Abstract-The method of caustics has been used for many years for studying deformations around crack tips in metals and other materials, but a systematic and easy-to-understand description of the principles and practice of photographing high quality caustics images has not been made available to potential practitioners of the method. This paper describes how sharp, high contrast caustic images can be obtained using simple equipment and procedures based on well-known principles of geometrical optics. The critical parameters are the light source diameter, which must be small, and its distance from the reflecting surface, which must be large; and high resolution, short depth of field and high initial magnification of the photographic system. Monochromatic filters or laser light sources can improve caustic images if the lens used is not highly corrected for chromatic errors. If collimated (parallel) light is desired to avoid making corrections for the divergence of the illumination, long focal length collimator lenses are necessary to minimize the light source. diameter contribution to the caustic ring thickness.
1. INTRODUCTION CAUSTICS are three-dimensional light figures in space, produced by reflection of incident light from plane polished opaque surfaces or refraction through plane transparent sheets having slight localized thr~-dimensional distortions (“dimples”) in the surfaces. The shapes and sizes of these figures, intercepted at a plane and revealed as two-dimensional, usually quasi-circular, line patterns, can be used to deduce the nature of the surface distortions giving rise to the figures. One class of distortions of special interest to investigators of fracture mechanics and crack growth in materials comprises the dimples produced around the tips of cracks under stress. The method of caustics has been used by several investigators for studying the state of stress in the vicinity of notches and crack tips, beginning with Manogg [l], with further development by others [2-71. However, the optical techniques by which high quality caustics are produced have been only sketchily described in the open literature. These techniques are not entirely obvious, and a great deal of trial and error experimentation is required to optimize them. Previous work [8] used procedures which, while adequate for the purpose, can be considerably improved upon using principles which are described herein. The discussion deals primarily with reflected virtual caustics, because these are the most useful for studying crack tip stresses in metals. 2. PHOTOGRAPHIC
OPTICS
Figure 1 shows a commonly recommended optical arrangement for photographing reflected virtual caustics (“virtual” because the caustic to be photographed does not exist in real space, but can be constructed from the reflected light rays by means of a lens). There are elements which either degrade the final image and serve no useful purpose (the screen) or are not absolutely essential and may degrade the final image if not carefully applied (the lenses in the light source path). Figure 2 is a schematic of the formation of a virtual caustic. Notice that the caustic is really a three-dimensional surface, and that the caustic image we are used to seeing is a two-dimensional section appearing as a bright ring, visualized in the transmission case by intercepting the real three-dimensional caustic surface at a particular plane via the screen, which can then be photographed at a suitable magnification with an ordinary camera. The use of a high quality screen in the case of a real caustic results in essentially zero depth of field, producing a very sharp, narrow caustic ring, assuming a suitable light source. In the case of the reflection virtual caustic, a real caustic image can only be formed on the screen with the help of a projection lens as shown in Fig. 1, and the depth of field and resolution of that lens will determine the sharpness of the caustic ring produced. Since one can directly project the caustic image at a suitable magnification on film in a camera without the intermediate screen, it serves no useful purpose except for visual EFM 43/2--D
185
186
D. A. MEYN Speciinen
I
d%z
‘Image Plane
Fig. 1. Diagram of a commonly recommended optical arrangement for photographing caustics [9].
reflected virtual
examination of the caustic prior to photographing it, and the viewing screen of the camera can be used for that purpose. The essential requirements of the camera optical system are high initial magnification of the caustic (on the camera film), large lens aperture and short camera lens-to-object distance, the object being the virtual (or real) caustic. The numerical aperture of the system and thus its resolution and freedom from aberrations is improved by the latter two characteristics. High initial magnification and large numerical aperture reduce the depth of field: a large depth of field would cause portions of the three-dimensional caustic surface in front of and behind the nominal object plane to be simultaneously in focus, thickening and blurring the caustic ring image. A small depth of field is therefore desirable, because it images a thin section of the caustic surface, producing a narrow caustic ring image. A high initial magnification also reduces the enlargement of the negative needed for a reasonable final printed image size, reducing the graininess of the print. Useful formulas for calculating numerical aperture, resolution, depth of field and magnifications from the usual lens and camera parameters, such as lens focal length andf/number (relative aperture) or lens aperture diameter, are presented in the Appendix. These simplified lens formulas assume that both principal planes coincide in the center of the lens, so that all measurements are made from the center of the lens (halfway between the front and back surfaces). These relationships may not be directly applicable for 35 mm camera lenses, which have highly unsymmetrical lens designs, but may be usable for many enlarging and large-format macrophotography lenses. When using these formulas, one should keep in mind that the theoretical diffraction-limited resolution is not ordinarily attainable with real lenses and should be used for lens comparison only. The depth of field values calculated are based on certain assumptions concerning visual acuity in a viewed image, and are merely indicative of how the caustic image may be degraded by increasing depth of field. Typical requirements for producing useful caustics images are a resolution of 5-10 pm, and a depth of field of l-2 mm. Such values are well within the capabilities of high quality macrocamera, enlarger, or 35 mm camera macro lenses having relative apertures of ,f/8 or better. LIGHT CONCENTf?ATlON t.AUSTIC)\
za
Fig. 2. Schematic of ray diagram for reflected virtual caustics [9].
Techniques of optical caustics photography
187
The following is a description of a simple procedure used to photograph caustics on fatigue-pre-cracked steel specimens in a high vacuum environmental chamber, which arose out of efforts to improve results obtained in earlier work [8]. The distance from the face plate of the vacuum chamber containing the specimen to the virtual caustic plane at 2, = 30 mm (Z, is the caustic-to-reflecting surface distance [2]), about 280 mm, dictated the required working distance, and therefore the focal length of the lens to be used. At this working distance a camera film plane magnification of x 2 was obtained using a 203 mm focal length f/8 achromatic doublet pair lens, with a long extension tube between the camera bellows and the lens, which was used wide open. Its theoretical resolution atf/8 is 5.3 pm at x 2, and tests demonstrated an edge-to-edge resolution of l&15 pm. The arrangement is shown schematically in Fig. 3. The angular separation between the camera’s optic axis and the light source of about 5.5” did not cause visible distortion of the caustic image. This system, with a suitable illumination source as discussed later, produced caustics of good quality (Fig. 4a). Subsequent resolution tests showed that had the lens been stopped down to f/16, higher contrast, better defined images could have been obtained without loss of real resolution, although depth of field would have been increased. Given the practical inaccessibility of the object, i.e. a virtual caustic “behind” the opaque steel specimen in a high vacuum chamber, the following method was used to set photographic magnification and Z,. The camera system was assembled on a track mounted on a sturdy tripod. The system was aimed at an accurately made ruler, and the ruler image was focussed and set to exactly x 2 magnification, with the help of another ruler placed at the film plane, the ruler and image simultaneously in focus in a magnifying viewer placed at the camera back. The bellows adjustment was locked, and never disturbed again, so that the image, when focussed in the camera viewfinder, was always at x 2. The camera system was then aimed at the test specimen, and the entire camera assembly was racked back and forth on its track until focussed on scratches or other defects on the highly polished specimen surface. A separate light source provided oblique illumination for focussing, as it was impossible to focus using the caustic-forming light source. The track was then racked forward exactly the distance Z,,, so that the camera, still set for x 2 magnification, was focussed a distance Z,, behind the specimen’s reflecting surface. The precision to which Z, can be set is related to the depth of field, which in this case was nominally 2 mm. Still better optical results could be obtained by means of a lens of larger aperture, or, given shorter working distances, a shorter focal length lens. For example, a 35 mm camera “macro” lens of 50 mm focal length with f/3.5 relative aperture working at x 2 (object distance about 75 mm) could theoretically produce 3.5 pm resolution and 1 mm nominal depth of field if optimally corrected for wide open aperture. Such lenses, because of their asymmetric design, may not require a long bellows to produce high magnifications, and they are corrected for short object distance. High quality, short-to-medium focal length enlarging or macro-photography lenses, corrected for short object distances, are also suitable for caustics photography, although they require longer bellows for similar magnifications. Note that ordinary 35 mm camera lenses may not be suited for caustics photography because they are normally corrected for large object distance, although they can be mounted backwards to circumvent this shortcoming. Lenses of large relative aperture (e.g. any/l .4 50 mm lens) are unlikely to approach their theoretical resolution because of compromises
LIGHT SOURCE
REFLECTING PLANE .
L
* CAMERA
di
VIRTUAL IMAGE PLANE
do-
LENS
Fig. 3. Diagram of a simple, practical arrangement for photographing
reflected virtual caustics. Note that the camera film plane is at the location of the screen in Fig. I.
D. A. MEYN
188
in their design. The addition of an ocular lens to magnify the image produced by the first (objective) lens would enable photographic magnifications of x 10 or more to be easily obtained, reducing the depth of field accordingly, and facilitate printing enlargement of even very small caustics. It is necessary to avoid too high a magnification, which can give rise to undesirable diffraction effects in the image [6]. 3. LIGHT SOURCE
REQUIREMENTS
The nature of the caustic-forming light source is of considerable importance in producing high quality caustics, but it is not necessary to use expensive laser sources. The author uses a fiber optic light pipe attached to a commercially-available fiber optic illuminator, which has a miniature high-intensity lamp with built-in focussing reflector, of the kind used in 35 mm slide projectors, to illuminate the light pipe. The focusing reflector serves as a condenser lens to demagnify the source, and the final section of light pipe, about 0.67 mm in diameter, serves as a secondary pinhole source. The most important characteristics of a good source are brightness and small effective aperture subtended at the reflecting surface. The latter requirement can be appreciated with reference to Fig. 5, which depicts the relationship between the light source diameter/distance ratio (its relative aperture) and its contribution to the thickness (and hence precision) of the caustic ring. By simple geometry, the contribution. t, of the light source relative aperture, d/L, to the caustic ring thickness is: t = Z,(d/L).
(11
Experiment has demonstrated that d/L should be less than 0.001 for the optics used for Fig. 4a so that the light source contribution will not significantly thicken the ring compared to other sources of blurring. Figure 4 shows examples of this effect for small and large d/L: and Fig. 6 is a plot of thickness measurements as a function of d/L, which shows that further improvement is not obtained for d/L < 0.001 for the camera optics used. The discrepancy between the measured results and the calculated value of thickness is probably due to the contribution of resolution and depth of field. If much shorter focal length, larger aperture systems are used at higher initial magnification, then a much smaller diameter high brightness source would be needed to realize the full benefits of such an optical system. This could be obtained by use of stronger condenser lenses and a smaller final pinhole, and in this case a laser might be useful, because of its high intensity, monochromatic output. Collimator lenses in the light path may be used to render the illumination parallel, to avoid the necessity and uncertainty of applying a correction factor [2] for divergence of the light from the source. The source would then necessarily be at the real focal point of the collimator lens, and the value of L in eq. (1) would then be equal to the focal length f of the collimator. In this case.
LIGHT SOURCE
REFLEdTlNG SURFACE
VIRTUAL IMAGE PLANE
d = SOURCE DIAMETER L = SOURCE-SURFACE DISTANCE 20 = SURFACE-IMAGE DISTANCE At = d&A, LIGHT SOURCE CONTRIBUTION TO CAUSTIC RING THICKNESS Fig. 5. Simplified diagram for deriving the contribution of light source diameter/distance ratio, d/L, to the apparent thickness of the caustic ring.
0
2 Source
4 Dia./Source
8
6 Dist.,
d/L
x1000
Fig. 6. Plot showing experimentally measured caustic ring thicknesses from photographs such as Fig. 4, versus normalized d/L. The solid line is the predicted contribution derived in Fig. 5.
10
Techniques of optical caustics photography
Fig. 4. Examples of caustics photographed as shown in Fig. 3. All photos taken using 4340 steel at K,= 62 MPa ,,/m, Z, = 30 mm, camera magnification x 2, with white light. (a) d/L= 0.0008, x 20. (b) d/L= 0.0021, x20.(c)d/L=0.0066, x20.
Fig. 8. Effect of monochromatic filtering on caustic images. K,= 44MPa ,/m, 2, = 30mm, d/L= 0.0016, initial magnification x 1. (a) White light, x 25. (b) Green interference filter (500 nm), x 25.
189
191
Techniquesof optical causticsphotography
q
976.2
A 9538
0
-
0.15 0.2 0.25 0.3 0.05 0.1 Zo/L, Relative iliuminati~n Divergence
Fig. 7. Plot of experimentally obtained caustic diameter ratios, D/Z+, as a function of illumination divergence factor, Z,/L. The solid line is the function Y’.~,eq. (2).
a lens of sufficientiy long focal length to satisfy the requirement for small d/L( = d/f) must be used to avoid excessive thickening of the caustic ring. If the source distance, L, is large relative to the originating circle radius, r, [2], the effect of divergence on the caustic diameter is small, and a collimator lens may not be needed for many types of caustics studies. For the virtual caustic generated under plane stress conditions at a crack tip the correction for divergence is approximately: M = D~“/~~~ = 1o.6= [(L - z@)/L]“~6
(2)
where Ddivand Spar are the caustic diameters for divergent and parallel illumination, respectively, and the expression y”.6 is derived from results of Theocaris [2f for caustics, replacing (L + Z&L (real caustic) with (L - Z,)/L (virtual caustic), and equating L = Zj in Theocaris’ nomencIature_ Given the caustic of Fig. 4a for which Z. = 30 mm and L = 837 mm, the angle of illumination on the specimen ro/L = 0.67/837 = 78 x 10m4 radians = 0.046O. Such as this divergence angle is, it leads to a correction factor M = 0.98, which means the caustic is about 2% smaller than if the illumination had been paraltef (co&mated). Figure 7 shows some experimental data for small Z,/L which provides a limited verification of eq. (2). Caustics in Fig. 4 were photographed using white light. Experiments were conducted with a green interference filter (wavelength 550 nm) in front of the illuminator to see if si~ifi~nt improvement could be obtained using monochromatic light. Figure 8 (printed in negative contrast because enlargement was done in a microscope) shows the same caustic photographed with white and monochromatic green light. Despite the marginal relative source diameter (d/L = 0.0016) and initial magnification (x 1) used, this illustrates that the caustic is better defined with monochromatic light. Evidently, the lens used has residual chromatic aberrations which degraded its performance in white light. A modern, highly corrected lens should show little improvement in image quality with monochromatic light.
4. PHOTOGRAPHIC
FILM
Experiments with various film types led to the use of Kodak Technical Pan 4215, processed for fine grain and wide tonal range in Technidol LC developer. This resulted in a working speed of ASA 25, requiring exposures of l/4-1/2 second with the f/8 lens at x 2 and the 0.67 mm diameter fiber optic ill~inator described above. Results with faster or higher contrast films were discouraging, resulting in grainy prints, and, because of more limited latitude, poorly defined ring structures. Obviously, however, if the photographic set-up uses a very high initial magnification with a large format camera such as is normally used for macrophotography, fiim requirements, except for wide tonal range, would not be so stringent, because less enlargement of the negative would be required in printing. In the present case, enlargements of x 10 to x 80 were used, depending on the size of the caustic image of the negative.
192
D. A. MEYN Table 1. Comparison of experimental and calculated caustic Darameters Caustic diameter
Calculated thickness values
dlL
Mag.
D,
Do
Ave.
0.008 0.008 0.008 0.008
0.25 2.0 0.25 2.00
0.80 0.75 1.35 1.39
1.10 0.865 1.75 1.50
0.95 0.81 1.55
1.45
Z,,= 30 mm. K, = 30 MPa ,J& (1st group),
5. MEASUREMENT
K, =
Meas. thick.
___ DOF
Resol.
d/L
Total
0.15 0.06 0.20 0.055
0.26 0.008 0.41 0.015
0.027 0.008 0.027 0.008
0.025 0.025 0.025 0.025
0.31 0.04 0.46 0.048
62 MPa fi
OF CAUSTICS
(30, + Do i. 4 0.875 0.77Y
I .45 1.J?.
(2nd group). Dimensions in millimeters.
AND
MAGNIFICATION
EFFECTS
The advantage of using high resolution optics at high camera magnification to obtain well-delineated, high-quality caustic images has already been discussed, but another, unexpected effect of camera magnification was observed. Table 1 shows measurements of caustics photographed under identical conditions except at different camera magnifications. Notice the two groups of data comprising two rows each; data within a group were obtained under identical conditions except for initial camera magnification. In both groups the average caustic diameter (measured as explained below) is slightly larger at the lower magnification: it is about 23% larger at x 0.25 than at x 2.0 for stress intensity K, = 30 MPa fi (small caustic), and 10% larger for K, = 62 MPa \,& (large caustic). The caustic diameter is conventionally measured at the midpoint of the ring thickness, along a line perpendicular to the crack line (vertically in Fig. 4). Detailed comparisons of the caustics photographed at these two magnifications showed that the increase in thickness at lower magnification was apparently asymmetrical, leading to a much larger increase of the outer diameter than of the inner diameter. The average of these two is the midpoint diameter, tabulated in the fifth column. The thickening of the caustic ring with decrease in magnification can be explained by noting that the depth of field, DOF, increases as magnification is decreased. The consequent thickening of the ring owing to increase in DOF, AfDoF, can be estimated from a plot of caustic diameter vs Z,,, with the ADOF set equal to AZ,, and ArDoF being read as AD,,“,,i,/2 (the change in caustic radius corresponding to AZ,). Table 1 presents data of this kind, and it is evident that the contribution of DOF to ring thickness is considerable at x 0.25 and accounts for the overall thickening with decreasing magnification. However, although the depths of field for the object distances experienced in this case are slightly asymmetric. the asymmetry is far from that needed to explain the asymmetry in the thickening of the caustic ring noted above. (Although the quantities calculated for DOF are useful for highlighting the contribution of DOF to apparent thickening of the caustic ring, one must keep in mind that the numerical values obtained are dependent on the criterion used to estimate DOF. The circle of confusion assumed in this case was 0.25 mm referred to the object at the initial photographic modification; it is not clear what this should be for a luminous three-dimensional light figure, for which the purpose is to quantify the extent of the caustic fore and aft of the object point contributing light to the image). The linear first order geometrical optics analyses developed for caustics formed at crack tips predict that the light intensity rises to infinity at the edge of the shadow spot, and abruptly drops to zero inside that locus [2,4]. Kamath and Kim [6] have shown by wave-optical analysis that the intensity changes in this region are more diffuse than that, and that the actual position of the geometric caustic point within the caustic ring, which is the first of several bright fringes (Fig. 4a) outside the shadow spot, is ambiguous. They showed by direct scanning of the light intensity of the caustic that the geometric caustic point occurred about midway between the inner edge and the point of peak intensity (Fig. 9). They point out that if the caustic diameter is used to estimate K,, overestimates will result from conventional caustic measurements, usually taken at the midthickness (average diameter) of the first fringe. Such overestimates will be greater for smaller caustics. Their analysis suggests that a practical measure of caustic diameter is the weighted average of the inside and outside diameters (0,and Do, respectively) of the first fringe: D,, = (30, + Do)/4
Techniques of optical caustics photo~aphy
1.2
1.3 Distance
1.4 1.5 1.6 From Caustic Center, mm
193
1.7
Fig. 9. Plot of the calculated third order optics light intensity distribution in the caustic [6] for 4340 steel, 11.6 mm thick, Z,, = 30 mm, K, = 62 MPa ,/m. The central shadow region is at left, and the first peak is the main bright ring of the caustic. The “caustic point” is the first order geometric orbits locus of the caustic.
where D, is the diameter at the caustic point. The last column of Table 1 shows DcPfor the present results; this concept does to some extent account for the asymmetry of the caustic ring thickening observed with decreasing ma~ifi~tion. 6. SUMMARY Experience with the photography of reflected virtual caustics from crack tips in polished metal fracture test specimens has demonstrated that acceptable caustic images can be obtained using a simple arrangement consisting of a 35 mm camera coupled by bellows and extension tube to an f/S lens, and an ordinary high intensity lamp~refl~tor system, coupled to a small diameter light pipe serving as the effective light source. The critical parameters are the final light pipe diameter, which must be small, and its distance from the reflecting surface, which must be large; and high resolution, short depth of field and high initial magnification of the photographic system. Monochromatic filters or laser light sources can improve caustic images if the lens used is not highly corrected for chromatic errors. If collimated (parallel) light is desired to avoid making corrections for the divergence of the illumination, long focal length collimator lenses are necessary to minimize the light source diameter ~ont~bution to the caustic ring thickness. Acknowledgemenrs-Thanks are due to T. W. Webb (Michigan Technological Univ.), E. Olfky and J. M. Lowry (Geocenters, Inc.) for experimental assistance, and to Prof. E. C. Aifantis (Michigan Technological Univ.) and Prof. E. I. Meletis (LSU) for their advice and encouragement.
REFERENCES III P. Manogg, Anwendungen der Schattenoptik zur Unte~uchung des Zerreissvorgags von Platten. Dissertation, The University of Freiburg, Germany (1964).
121P. S. Theocaris, Elastic stress intensity factors evaluated by caustics, in Mechanics ofFracture (Edited by G. C. Sih),
pp. 189-252. Martinus Nijhoff, The Netherlands (1981). ]31 J. Beinert and J. F. Kalthoff, Experimental determination of dynamic stress intensity factors by shadow patterns, in Mechanics of Fracture (Edited by G. C. Sih), pp. 281-330. Martinus Nijhoff, The Netherlands (1981). I41 A. J. Rosakis and A. T. Zehnder, On the method of caustics: an exact analysis based on geometrical optics. J. EZusricity 15, 345-367 (1985). [51 R. Hermann and N. J. H. Holroyd, Environment sensitive fracture of AA7475 using shadow optical method of caustics. Mater. Sci. Technol. 2, 1238-1244 (I986). spot of a crack under mode I loading: theory and experiment. I61 S. M. Kamath and K. S. Kim, Coherent-lint-shadow Exp. Meek. 26, 386-393 (Dec. 1986). ]71 E. E. Gdoutos and E. C. Aifantis, The method of caustics in environmental cracking. Engng Fracture Mech. 23,423-430 (1986). 181 D. A. Meyn, T. W. Webb and E. C. Aifantis, Hydrogen-assisted cracking studies of 4340 steel by using the optical method of caustics. Engns Fracfwe Meek 33, 913-925 (1989). ]91 A. T. Zehnder, A. J. Rosakis and R. Narasimhan, Measurement of the J integral with caustics: an experimental and numerical investigation. SM Report 868, Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, U.S.A. (1986).
194
D. A. MEYN
APPENDIX: 1. The following m = f= F = D = d, = do = DOF = I = n = na. =
conventions
RUDIMENTARY
OPTICS
FORMULAS
are used:
linear magnification at image plane lens focal length relative aperture or f‘/stop number, i.e. the “8” in ./‘/I. aperture diameter of the lens =f/F image distance from lens second principal plane object distance from lens first principal plane depth of field, total distance on both sides of object plane over which wavelength of light index of refraction = 1 in air and vacuum numerical aperture.
image is in satisfactory
focus
(Note that first and second principal planes coincide at the center of a simple lens and to a good approximatton of some symmetrical compound lenses.) 2. The following formulas apply in all cases. For simple lenses and some symmetrical compound lenses. especially those of longf, measurements can be made from the midpoint of the lens thickness, otherwise measurements are made from the principal planes: m = d,/d,, l/f= Assuming
l/d,+
l:d,
D 6 do so that 1 = tan a and n = 1 for air: n.a. = D/(Zd,) =/‘/(Zd,F). Resolution
Letting
E. = 5.51 x IO-“mm Resolution
(green
= 0.61EJn.a.
(theoretical
diffraction
limit).
light):
z 6.72 x 10 ‘do/D = 6.72 x 10-4doF/‘rr DOF = d,/(3mD)
6.72 x IO-“(m
+ I)Fim
z (m + I)F/(3m’).
Ordinarily, “m” includes printing enlargement for DOF, but for caustics, “WI” is taken as the initial camera magnification. This formulation assumes visual acuity of 0.167 mm, and theoretical resolution < DOF. 3. For example, consider the 203 mm, .f/S lens discussed, operating at a camera film (image) plane magnification of x 28: do =.f(l + l/m) = l.Sf=
304.5 mm
d, = md, = 609 mm D =,f/F
= 25.4mm
n.a.,,, = D/(2d,) Resolution
zz 0.61E.in.a.
= 0.0417
= 3.36 x 10-4/n.a.
DOF z d,/(3mD)
= 0.00806 mm = 8.06 pm
= 2.0 mm.
(Received 28 August 1991)