FIG. 1. "Short" system designed by MATRA for photogrammetric automation.

DR B. L. Y. DUBUISSON20 Avenue Paul Appell

75014 Paris, France

Automatic Photog ram metricCartographyBecause of the arduous and painstaking nature ofcartographic drafting and the scarcity of those highly trainedartisans, together with improvements in applicable computerhardware and software, the time has arrived for theapplication of specially programmed digital drawingmachines.

(Abstract on next page)

DIGITAL DRAWINGS-WHY MAKE THEM? HowToMAKE THEM?

D IGITAL DRAWINGS AND MAPS are establishedafter determining the values of the coor­

dinates of the points composing them. Theirorigin can be traced to the very start of geo­graphic sciences because, except for the plotsmade on the surveyor's table and by graphi­cal stereoplotting, all cartographic drawingsare composed of conventional representa­tions on points defined by coordinates.

From the beginning, large-scale maps of

surveyors are based on calculations in coor­dinates serving to define the perimeters andthe surface-areas, and then finally to plot thegraphical drawing that forms the definitivedocument. Of course, these digital drawingsand maps comprise a non-negligible part ofgraphical fllling in.

Recently, however, the number of graphi­cal plotting processes still employed fell offsharply with the advent of new topometricfacilities and computers. In this connectionmention should be made of the general au-

65

66 PHOTOGRAMMETRIC ENGINEERI IG & REMOTE SENSING, 1975

tomation prospects offered by instrumentsfor the electromagnetic recording of anglesand distances, such as the A. G. A. 700 andthe generalization of the presence of mini­computers.

In the photogrammetry itself, the digitiza­tion methods have been developed throughthe hequent lise of detectors in the modelsupplying, by digital-numerical transforma­tion, the coordinates measured in this modelspace. This development, it should be noted,has been caused by the development ofanalog aerotriangulation and the require­ments ofcivil engineering. But we are still huremoved, at this level, from what may becalled a digital draWing, in its present con­ception.

Today one may define a drawing or an in­tegral digital map "as a map which is solelyformed of alpha-numerical data, classed infiles, and of a nature to supply automaticallythe partial or total expression of the space atan arbitrarily fixed scale".

trolling the drawing may be prepared withconventional instructions linked to thescale of this drawing, and this latter may bemade at the desired graphical dimension onthe drawing machine. What is important isthat the same address flIes of coordinatesand instructions may serve as a base fordrawings and maps on all scales and at allinstants, especially if the card Hies are keptupdated with the modifications embodiedin the meantime. This independence, ex­tremely valuable, is obviously only relativebecause the numerical accuracy of thecoordinates and the superficial density ofthe data, once the acquisition has been per­formed, are invariants of the system. Thereis therefore an upper limit to the possiblegraphical drawing scale.

• The drawings and maps are always in­tended to allow studies, or to make deci­sions in a certain well-defined political,economic or technical field. This is why theuniversal graphical maps are rarely usedstraight off the board. The utilizer makes aselective extract, prior to processing it; thisforms a thematic map, with unnecessary

ABSTHACT, Some general remarks on digital drawings and the devicesthat have had to be created to produce a trullj complete automation ofpllOtogrammetric maps. Description ofdigital Sljstems ofcoded dataacquisitions; adaptations performed on various tljpes of optico­mechanical instruments; sljstems grouped together in complex ac­quisition sljstems. After three years operating several sljstems, thefindings are given on the extreme modularitlj of automatic photo­grammetrlj and some forecasts are made on the phases of its futureexpansion.

This definition implies the need to knowthe points of the tracing in orthogonal geo­graphic coordinates (X, Y, Z), and to associatewith each point two items of information: theform code for the tracing of the lines, and thefunction code, both for the conventional ex­pression required for the drawing and for theclassification of types of objects.

The advantages of such "integral" digitaldrawings are considerable; to state these ad­vantages amounts to mentioning the veryeconomic requirements which have entailedtheir creation. One should note that:

• The value of the stored data is integrallypreserved because its preservation does notdepend on a graphical transfer (which isalways imperfect) made on a support gen­erally unstable with respect to time.

• The technological developments in hard­ware and software allow a complete auto­mation, right up to the graphical expression,if the data acquisition is digital.

• The relative independence vis-a-vis thescale of the graphical expression which willbe most often used. The magnetic tape con-

and cluttering details removed. The greatutility of the complete digital drawing re­sides therein; one may extract from it, whenneeded and in accordance with the utilizersrequirements, the data selected by compu­ter and drawn automatically in thematicform.

• Information contained in the large collec­tive card flies, such as that dealing with thepopulation, the economy, industry, fiscalmatters, must be located geographically.Thus, in all countries of the world, a needhas existed for digitized location data proc­essing which only digitized drawings cansatisfy.

• Finally, the topographic and cartographicdrawing is an arduous and costly work ofanarti stic nature, performed by specialistswhose number and qualifications are tend­ing to fall off. The time has therefore comefor this work to be taken over by speciallyprogrammed drawing machines. This canonly be done through the digital drawing.

~IETHODOF ESTABLISHING DIGITAL ~lAPS

Maps are made by land surveys and resultfrom the topometry, or else by aerial photog-

AUTOMATIC PHOTOGRAMMETRIC CARTOGRAPHY 67

raphy. One should distinguish first of all themaps and drawings plotted directly in thedigital mode, and recorded as such with theircodifIcations, allowing the subsequent clas­sification in a card file, and the selection in acomputer prior to their drawing.

Next may be distinguished those maps al­ready drawn conventionally in the graph icalmode; the graphical plotting of the cOOl·di­nates ofthe points and of the required codifi­cation is done later. This is done, of course,with digitizing tables in X and Y, equippedwith their peripheral units. Such a practiceinvevitably entails a loss of data because aprevious passage by graphical channel, hastaken place.

Prior to the advent ofdigital systems (ea rly1971), the possibi Ii ties of digital photogram­metric plottings were offered by the follow­ing hlcilities:

a) analytical plotters (APC), perfectlysuitable, through their principles, for dealingwith the trend to integral digital methods. Forvarious reasons, they have not been instrumen­tal in achieving an integral automation, trulyoperational and industrially productive.b) recording systems developed in numerousphotogrammetry optical-mechanical instru­ments, currently available. The system is slow,firstly because it is necessary to perform manu­ally the marking of the points taken on a as­sociated document, i.e., a photograph. Next,and above all, as the system only supplies thecoordinates in the machine space, and becauseone must then transform them oH~line intogeographical coordinates by transfer with acomputer unit separate from the restitutionworkshop. The result is a break in the produc­tion continuity, entailing technological difficul­ties, and responsible for low productivity.

The general and widespread use of digitalphotogrammetry, and its orientation towardsautomatic photogrammetry have thereforeled to a complete re-evaluation of the system.

DIGITAL SYSTEMS IN SERVICE

One system, called shortsystem, has beendesigned by Matra for photogrammetric au­tomation (Figure 1). The program proposedwithin the concept of the digital system ofphotogrammetric acquisition comprises fIvemain phases:

• Performs of the measurements at a highrate, simultaneously with the encoding ofthe data for their subsequent classing, in­cluding the indications for the preparationof the maps, such as the toponymy.

• Transforms the machine coordinates intoorthogonal geographical coordinates andtheir recording with the encoding on a fast

memory, by on-Ii ne association with a sys­tem and a mini-computer.

• Ensures the material nature of the pointsrecorded in the memory, thanks to an in­stantaneous transfer under the eye of theplotter. The points extracted are marked forexample, on a pre-established 01'­

thophotomap, which system is calledorthoguidance.

• Performs checks on-line through loopingon the different actions of the plotter, bysignaling by optical, graphical or soundmethods.

• Creates finally ofa primary magnetictape which is compatible with the sub­sequent processings, called downstream,of the map automation operation. Thisdownstream processing system may beeither a computer and large automaticdrawing center (e.g., of a public commun­ity) or a system annex, but separate from theupstream system mentioned, and workingofl~line with it. The role ofthis downstreamsystem is to classify the information follow­ing the subsequent utilization (geographicclassing or classing by subject), to createdrawing instruction tapes and to producethe automatic drawing. This downstreamloop will not be analyzed in greater detail,it is not especially photogrammetric, butapplies to all automatic cartography.

A block diagram showing the main generalorganizational functions of the short digitalsystem is shown in Figure 2.

It is certainly true that other configurationsmay be conceived. They may be simpler; atthe limit, these configurations amount tooptica-mechanical instruments equippedwith a recording of machine coordinates onpunched tape, of low efficiency, and whichhave been mentioned above. Other struc­tures may also be more elaborate culminatingin self-correcting systems complete withtypewriter, forming autonomous data acquis­ition and processing systems, a description ofwhich is given next.

It should be emphasized that this type ofsimple or short system, described before isdesigned to feed a center of data and drawingbanks; from this very fact its use is limited tothe acquisition and the pre-processing ofdata. The production of several similar sys­tems is coordinated with that of a single databank center, which assures the processingand the drawing. But, in other instances, it isperfectly in order if, from a study of the cost­production parameter, preference is given tocomplete self-correcting systems, called longsystems.

No rules are laid down as to the type ofstereoplotting instrument, in any of the con­figurations described. The first four systemscreated have been adapted to "MATRA type

68 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975

Classing by geographic referenceunits of processing

Coded magnetic recording of points andshapes in orthogonal geographic coordinates

"Downstream It processi ngloop of the acquisition

system af automaticcartogra h (1)

Dota classedin accordance with

the disci pi ines

Thematic mops onpartial regions onthe whole region

Flc. 2. General functional block diagram-sample of a "short" system.

Functions of the digitol acquisition loop

Stereoscopic plats recordedautomoticall

Encoding of linesand ob'ects in telet

G rophicol check of plate recordedon orthophatamap (or other docu­

ment). called "orthoguidance"

Computer check of procedure

Calculations for scaling and fortransformat i on of "machine eoordi­notes" into II eo ro hie coordinates"

Primary magnetictape of mopping

Q

FIC. 3. Block diagram ofthe functions ofthe digital acquisition loop-sample ofa "short" system.

AUTOMATIC PHOTOGRAMMETRIC CARTOGRAPHY 69

;k I ~InterlaceComputer ~

Magnetic4K tape

0' recorderlTeletype ond 8K

0 0punched lopejy o 0 a 0 0 0

Functions of elements--~-_._----

Associations of

the magnetic tapewi th differentcounters (of cha­

racters, etc.)allowing the work

of the subsequentprocessing compu­ter to be minimiz­

ed.

Display of Z

Checks of format

and of recording.Display of X, Y rules.

·Coding of PJIses Conversion of Recordings com­emmitted by the "model coordina- potible with thedetectors. tes" into t1geogro- subsequent pro­

phic coordinates" cessing computerfor the drawingof ~ops and theclassing in dotocard fills.

Introduction ofcomplementarydata and coordi­notes.

Instructions to t.hecomptuer.Resu I ts of col cu­lotio(\~ and agree-ment o-f:pointS.'~ Scale changing

Error de,tection devi ce.signol.

Digitizin~ ahd,wr.iting of.,the

toponymy.

Acquisition bystereoscopic plott­ing.Or XY mopping ona document on a

luminous table.

Check of acquisi­tion in the model

by marking on a

document placedon a "orthogui­dance" table.

FIG. 4. Block diagram of the "short" system configuration of digital acquisi­tion (made by MATRA-SFOM).

920" projection stereopolotters. But othershave been made from Wild, Zeiss and Kernoptical-mechanical instruments.

A good method ofchoosing the type ofplot­ting instrument is to highlight the respectivemerits and drawbacks of each one.

The projection plotter (Figure 5), speciallyfitted out, has the following good features: (a)it only requires one operator, (b) it takes ad­vantage of the instrument's luminous table to

FIG. 5. A projection-type plotting instrument.

organize, on the orthophotomap or the draw­ing, the acquisition program, the placing ofthe drawing in sheets, the toponymy, etc.,and to follow directly on this table the check­ing ofthe recorded points, by ortho-guidance(or plano-guidance), (c) it completes themeasurements of the model by recordings, ifnecessary in the XY plane (with an adaptedmagnifying glass), to introduce numericallyin the recording, after affine conversion, ifnecessary, and (d) it uses occasionally thesystem for the X, Y recording of existingdrawings. However, it has drawbacks: (a)lack ofcomfort for the operator, (b)limitationsof model enlargement, and (c) an accuracy alittle poorer than that of optical-mechanicalinstruments. '

The advantages of the optical-mechanicalplotter (Figures 6 and 7) are: (a) higher accu­racy (all this greater precision may be usedwithout any failure by the digital system); (b)offers good comfort, (c) use widespread innumerous countries, which leads it quitenaturally to be considered as forming a com­plement to digital systems. Drawbacks of theoptical-mechanical plotter are: (a) the dis­tance at which the operator is located fromthe ortho-or plano-guidance table (but thisdefect may be remedied, luckily, by the addi­tion of a television system, detecting theimage at tracer level and reproducing it on a

70 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975

FIG. 6. An optical mechanical plotter.

FIG. 7. Additional optical mechanical plotters.

screen in front of the operator), (b) highercost, entailing higher capital investment.This shows that the type of plotting instru­ment is independent of the digital systemused (and vice versa?).

Let us consider some of the details of themodifications made (for the first four units) inshort digital systems on projection plotters.

On the instrument, a precision ~oord­inatograph has been adapted to the table. Itcomprises a rotary pulse detector for each ofthe X and Y movements. The carriage of themobile arm (rail of the X coordinate) isequipped with: a mobile screen in Z withpulse detector, provided with a stereoscopicpointer index; a pricker-pencil moved by thecoordinate recording contractor, the action ofwhich may be engaged in a variety ofways; acontrol by electric motor ofthe screen's verti­cal displacement; the usual accessories forthese instruments, such as the stereoimage

alternator; a retractable magnifying glass al­lowing X, Y acquisitions to be made in theplane of the table; and the X, Y, Z recordingpush-button.

The teletype (Figure 8) comprises a tapepunch; it enables one to dialog with the com­puter to which it is directly connected.

The interface, with X, Y, Z luminous dis­play (with pre-selection knob), allows aboveall the encoding of the pulses emitted by thedetectors of each of the coordinates, and theassimilation by the computer of coordinateconversions. It carries: a scale changer, forreading of Z with pre-selection; and an au­tomatic trigger on time constant, anothertrigger on distance constants, and a third ondistance constant of X and Y.

The computer (Figure 9) has a capacity of4K or (better stilI) of8K words. Its programs,introduced rapidly by magnetic tape, are evi­dently specific for the type of work foreseen.

AUTOMATIC PHOTOGRAMMETRIC CARTOGRAPHY 71

FIG. 8. A teletype station. See also Figure 10.

FIG. 9. Tape drives and computer.

The role assigned to it: allow the rapid scal­ing of the photogrammetric model with cal­culation and storage of the coordinate con­version parameters (which may be conformor affine); convert the model coordinates intogeographic coordinates; assure the largestpossible number of recording checks-orderin which the instructions are introduced (thefiling of data by map edition sheet differs, inactual fact, from the filing of data extractedpair by pair); check of the omissions in thedrawing of filing instructions; check of obvi­ous incompatibilities, etc...These checksare interspersed with response instructionsgiven to the teletype of the operator. Thisset-up relieves the data bank computationcenter and above all renders this band au­tonomous of the "data acquisition systems".

Already, a large number of programs havebeen compiled, e.g., for stereophotogram­metric plotti ng ofurban maps, for tracing con­tours, for topographic surveys of urban vol­umes on a very large scale, and the restitutionoftheir perspectives ofthe most varied tapes,and for the integration ofold maps in a photo­grammetric out-line, etc.

Ofcourse, each specific regulation ofa typeof survey leads to modifications of acquisi­tion and computer programs. This gives agood idea of the extreme flexibility and themodularity of the system.

The low cost of the system is assured by itshigh productivity rating. The number ofpoints carefully processed in an hour on theshort digital system, described above, variesfrom 280 to 400, depending on the geo­graphic difficulties and the operators. Thus,the cost of a point is less than a quarter of thecost resulting from the best recording de­vices most frequently used. But in addition tothis interesting feature, there is the economybrought about by the simultaneous creationof data card files, or banks, sorted by thecomputer of the downstream operatingcenter, and also that of the automatic drawingwhich concludes the operations.

DIGITAL DATA ACQUISITION SYSTEMS OF THE

AUTONOMOUS

LONG SELF-CORRECTING TYPE

In a general manner, the acquisition sys­tems such as those ofthe short system alreadydescribed, as well as those of self-correctingsystems of which mention will be madefurther on, have an autonomous structurewhich is of general application.

This signifies, in particular, that their con­figuration may be transposed in all the sys­tems ofautomatic cartography. This is true forexample for systems based on the utilizationof data derived directly from topometric sur­veys on the ground. This applies also to ac­quisitions on the X, Y digitizing table of ele­ments of a map already drawn.

This must be SO, in actual fact, to allow theadaptation to all the cases of cartographicwork organization. One may, for example,mention a controversial question, i.e., is it, oris it not, preferable to perform either (a) adirect digital acquisition in the photogram­metric model, completed or updated by a sys­tem of terminal able to dialog with the com­puter, or (b) a conventional graphical restitu­tion completed by data gathered on theground, and then digitized on the X, Y table.

The answer depends both on the tradi­tional habits of the organization performingthe work, and on the development and theutilization facility of the dialoging terminals,for the corrections and updatings. This lastpoint is finally linked to the performance ofprograms of organizing cartographic files.Present trends seem to indicate that the firstsolution will be the course followed in thefuture. Now, the long self-correcting systemmeets precisely this conversational require-

72 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975

STKIlEOPLOTTER

TELETYPE

o DISK

STOR.lGE

MAGNETIC

FIG. 10. Block diagram of the self-correcting system.

Power drive Power drive

NOVA COMPUTER

32K

8D;,k 1.2 M

Power drive

@mognetic

te e

@agnetic

• BK

Computer

FIG. 11. Block diagram of a system with a single on-line drafting machine for"intermediate" and "final" drawing.

ment of checking, correction and updating.The self-correcting system, designed by

the same specialist of systems as the "shortdigital systems" is intended to render theacquisition system independent of the sub­sequent data processing system. It assuresthat the primary magnetic tape shows neithererrors nor omissions, thanks to a graphicaloutput. It should also be stated that the effi­ciency with which this action is performedmay be made high enough for this output tocreate the final drawing. The role of the

downstream processing center, thus freed isin this instance only to perform the classifica­tions in selective card files intended to meetthe general and subsequent requirements. Inaddition, this system assures the uniquenessofstored coordinate values for the same pointbelonging to several different and associatedclassifications. A block diagram of theself-correcting system is indicated in Figure10.

This system is therefore more complexthan the short digital system because one

AUTOMATIC PHOTOGRAMMETRIC CARTOGRAPHY 73

must add, firstly, a drawing table and its as­sociated program and, secondly, a desk unitto perform the corrections and updatings. It isalso equipped with a proximity search soft­ware to reveal the identical points and to cal­culate mean values.

OPTICAL MECHA:\ICAL PLOTTERS

One coherent data acquisition and automat­ic drawing system groups three assembliesof digital systems associated with opticalmechanical stereoplotters. This forms a sig­niflcant example of the modularity of thesesystems allowing them to be adapted to thespecific requirements of a particular organi­zation. It consists of a MATRA system placedin service in a large national geographiccenter.

This ensemble comprises a groupe ofthreeacquisition stations, on Zeiss and Kern in­struments (Figures 11 and 12), processed intime-sharing by a Nova computer of32 K cen­tral memory, coupled to a disk store of1,200,000 words of memory, a teletype sys­tem and two IBM compatible magnetic tapedrives.

Each of the three data acquisition stationscomprises three optical encoders sensitive to1/100th mm, a MATRA interface comprisingthe display of the X, Y coordinates a multipli­cation factor of the Z recordi ngs, and a variety

of devices for the recording of the points, theautomatic incrementation of a point numberand flnally a hard copy output teletype, todialog with the computer. This system hasthe following lcorking program: acquisitionchain control; access blocking to give priorityto teletype system; acquisition checking;image point setting and various correspond­ing checks; calculation of the geographiccoordinates and various checks.

The principle of operation is as follows:each station records its data in the locationassigned to it in the disk; after checking, thetransfer is performed without any preferen­tial order of the disk on the tape; outside theacquisition periods, the data are rewritten onthe second magnetic tape after sorting on adisk (tape-to-tape work); this new tape is theflnal tape arranged by map sections-it thenserves to establish the control tape of thedrawing instrument.

Checks and corrections have been care­fully studied. In spite of the large number ofchecks performed by the acquisition pro­gram, the very exactness ofthe acquisition hasnot yet been tested, as it should be before thefinal drawing, while the photogrammetricpair of images is still in position. Thesechecks may be executed on a display con­sole, but it has been thought preferable to usea small drawing table to perform these

IElEJ

6magnetic

l- --I l~!,,e

g

FINAL DRAWING

disk 1.2. M

1-----1 NOVA COMPUTER 32 K

DATA CIRCULATION,

1..cquf.ltlon ..... 32 K ...... di.k2.dl.k~roll.r t.bl.~dl.k

(eheckino·corredlon)

OUTSIDE THE ACQUISITION,

magnetic tape 1~ dlak~ maanetlc tape 2:tape for off-line drawing

FIG. 12. An alternate version of Figure 11 consists of placing the drawing table in an areadifferent from that of the data-acquisition plotters.

74 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975

intermediate drawings (every 500 points, forexample). The acquisition errors discoveredmay next be directly corrected on a disk. Ofcourse, the final high-class drawing, sectionby section, is established in its integrity off­line. This arrangement is -shown in the dia­gram shown in Figure II.

Another version consists in only using asingle on-line drawing table (linked to the 32K computer, completed with the multiplexerof an 8 K computer with teletype) which per­forms its work without interfering with dataacquisition. The final drawing is then draftedoutside the acquisition area with the sameequipment and the 32K computer. The blockdiagram of this set-up is given in Figure 12.

FUTURE OF COMPUTER-PROCESSED NUMERICAL

PHOTOGRAMMETRY

What developments may one expect, in theshort term? In the first place, it is expectedthat existing stereoplotters, projection oroptical-mechanical, will be equipped withdigital acquisition systems. Grouping plot­ters in series, working in time-sharing, willbe greatly appreciated. This is, moreover, adevelopment which is already underway. Itis also well to stress that such systems are not,as one might be led to believe, created byassociating hardware units but, on the con­trary, involve compatibility and softwareproblems which are both arduous and dif­ficult to resolve, even for engineers specializ­ing in the creation of such systems.

In the meantime, one may expect first ofallthat developments will take place in theequipment of digital systems on existing

anlog instruments and probably, self­correcting and tracing autonomous systems.Next in this perspective, one may expect thatthe reliable analytical plotters will come intoservice; in fact, this automation channelshould open up an industrial activity field forthese instruments. One must admit that suchan opening was not clearly evident at thetime ofthe advent ofthe first analytical units.

In the long term, it is not unreasonable tobelieve in an extension of complete digitalsystems, but supported by data deriving fromreliable analytical plotters and often groupedin series.

In conclusion, it is quite evident that anincrease in cartography productivity will beachieved through automation. Inasmuch asthe greater part ofthe production ofdrawingsand maps is photogrammetric, it follows thatautomatic photogrammetric cartography ishenceforth a fundamental method.

It is perhaps worthwhile considering whatconcerns men as such. Emphasis must be laidon the development of the technologicalqualifications of the technicians responsiblefor the systems and the plotters; this factor,i.e., better skills, is fundamental-it con­cerns elementary statistics, programmingand electronics.

It is true that all countries now possesscertain details, drawn from experience, inthese fields to which those interested mayrefer if they so wish. This point should bewell-known, and stressed, to allow conven­tional photogrammetry to be placed in its cor­rect historical setting, and a new chapteropened, more flexible, more adaptable, be­cause it is more modular.

Errata

Several printing errors were included inthe article, "A Theorem in Least Squares," byH. S. Williams in the November 1972 issue ofPhotogrammetric Engineering. In Equation8 on page 1128, the fourth term should be

Ai' Wi f 1

and in the last equation on the same page, thefirst term should be

W2A2 Ql -1.

The second term of the equation at the top ofpage 1129, a subscriptl was omitted from thelast element:

Q-1A 2' W2 A 2Q1 -1.

The last term on the second line beneathEquation 13 should be an X instead of Q:

A 2 Q-1 (Q - QJ /U"' * .

In the second term of the second equationbeneath Equation 15, the asterisk after Dshould be deleted:

Q-1 A 21(A 2 M' + f,J .

In the third line from the bottom of the samepage, a plus sign should appear instead of aminus:

(A' M* + f) .and the last term of Equation 17 should be

-/U"*'Q1 ./U"* .On page 1130, the last term before Equation19 should be

+ X' *, Q1 . /U"'* .The Editor regrets these mistakes.