4
Anti-gravityelectronics A nti-gravitationalbehaviourofforce- precessedgyroscopesposesperplexing problemsforscientistswhoarewell versedintheprinciplesofNewtonian mechanics .Thesemachinesdemonstratea liftforcethathasnocounterpart,asrequired byNewton'sThirdLawofMotion,unlessone imaginesthatabalancingforceisexertedon theether . Traditionallywehavecometoacceptthat actionmusthebalancedbyanequaland oppositereaction .Itistruethatmaterial bodiesarecausedto move byout-of-balance forces,buttheythenaccelerateandthere reallyisabalanceforcerepresentedby inertialreaction .Whathasnowemergedon thetechnologicalsceneisamachinethat canproducethatout-of-balanceforcewith- outaccelerating .Thismeansthatitcan, whollyorpartially,offsettheforceofgravita- tionandsodefytheaccelerationeffectsof gravitation . Someinterestinthissubjecthasbeen shownintheFeedbackcolumnsof Electro- nicsandWirelessWorld (forexample,see January1987,March1987andAugust1988 issues) .Forthisreasonandthefurther reasonthatthephenomenonhasanexplana- tionintermsofelectroniceffects,the followingaccounthasbeenwrittenfor E&WWreaders . NEWTON'SRULE Studentsofphysics,whenconfrontedwith Newton'slawsofmotion,areledtoaccept thatwhenmatterinteractsbycollisionor otherwise (e.g. viaelectricorgravitational forces)thereisseparateconservationof linearmomentumandangularmomentum- Actionandreactionarebalancedandthis hascometomeanthatnoself-acting machinecandevelopapropulsiveforce withoutsheddingmassinsomeform-Simi- larly,wehavealwaysbelievedthatno machinehavingarotorandastatorcan developitsowninteractiontorotatethe rotoratconstantspeedwithoutapplyinga balancingtorqueonthestator . Textbooksthenarguefromthisaction- reactionlawandthelawofenergyconserva- tionthatwhentwoperfectlyelasticbodies collidesoastosuffernoenergylossbyheat or radiationtheymust comply withwhatis knownasNewton'srule . Thisrule,youwillremember,saysthat therelativevelocity of thebodiesafter ReinterpretationofNewton'sThirdLawofMotion suggeststhatitdependsuponanelectronicaction . Electronicinteractionthereforeexplainstheparadoxical anti-gravitypropertiesoftheforce-precessedgyroscope . H . ASPDEN THELAWOFELECTRODYNAMICS TheLorentzversionoftheelectrodynamicforcelawfortheforcecaused byaunitelectromagneticcharge movingatvelocityvactingonaunitelectromagneticchargemovingatvelocityv',separatedbyaunit vectordistancer,canbewrittenas[v' .[v.r]]invectorproductnotation .Whenthissameexpression is formulatedinscalarproducttermsitbecomes, (v' .r)v-(v.v')r Thisforceisnotbalancedwithreactionbecauseitdoesnotliealong rowingtothefirsttermandbecauseit changesmagnitudeifvandv'areinterchanged . ClerkMaxwellknewthataterns (v.r)v' couldbeaddedwithoutthisaffectingtheempiricaldata-Such a termimpartsasymmetrywhichassuresbalanceoflinearactionandreaction. butallowsanout-of-balance couple . Thealternative-whichcanbeshowntoaccountforinductioneffectswithenergyconservation,isto subtractsuchaterm- (v' .r)v--(v,r)v'-(v .v')r Thentheformulaassuresnoout-of-balance coupleandsoconservationofangularmomentum for electronicinteraction-Itgivesanout-of-balancelinear force,butitcansatisfytheformofgravitational interaction-Thisiseasilyseenforthesituationinwhichthe gravitationaleffectissetupbyfundamental char gesofthesamepolarity,samemassandsamevelocity (v-= v')- Thenthefirsttwotermscancelto leaveamutualforceofattraction actingdirectlybetweenthecharges andfullysatisfyingthe action-reactionlawofNewton . impactis - E timestherelativevelocityof thebodiesbeforeimpact .Here E iswhatis knownas'thecoefficientofrestitution', whichhasavalueofunityforperfectly elasticloss-freecollisions . Whatisneverexplainedintextbooksisthe chicken-and-eggtypeofquestion,namely : 'Whichcomesfirst,Newton'sThirdlawor Newton'srule?'Whydowetaketheaction- reactionlawasfundamentalandnotNew- ton'srule?IfNatureactuallydeterminesthat Newton'sruleisthemorefundamentalof thetwo,then,giventhatenergyiscon- served,wecandeducethatactionbalances reaction . Now,ofcourse,itisimmaterialtobother aboutNature'sprioritiesifboththeaction- reactionlawandtheruleareunquestionably validinanyphysicalsituation .However, havingdiscoveredthattheaction-reaction lawcanbebreached,thereispurposein wonderingwhetherNewton'sruleisan expressionofamorebasicfundamental truth . Thenitbecomespossibletosaythat, providedenergyassociatedwiththelinear, trans .[ationa .[motionoftheinteracting bodiesisconserved,therewillheconserva- tionoflinearmomentumandsobalanceof actionandreaction .However,thisargu- mentpermitsustoimaginethatsomeof thatenergycanbedrawnfromtherotary motionofoneofthebodies .Inthiscase,we willnotfindperfectbalanceofactionand reactionorconservationoflinearmornen- turn .Wewill,inthisspecialsituation,he abletounderstandhowaflywheelcanslow downwhilstusingitsenergytomovethe systemlinearlyagainsttheforceof gravita- tion . THEUNDERLYINGELECTRONICS TheimplicationfromthisisthatNewton's ruleisthemorefundamentalcharacteristic ofinteractionsbetweencollidingorinteract- ingbodies .Flowcanelectronicsbeinvolved? Well,letusnotrestrictthemeaningof electronicstotheflowofelectroncurrents incircuits .Electronicactioncanhethatof theatomicelectronsbroughtintocollision withthebodies . Considertwoequalchargesofthesame polarityandimaginethattheymovealonga commonlinesoastocomeintocollision . Theirrelativevelocityisameasureofthe mutualelectromagneticfieldinthenear vicinityofthecollision .Theenergyinthe fieldatthemomentofcollisionisprop- ortionaltothatrelativevelocitysquared . Energyisconservedinthecollision .There- fore,immediatelyafterthecollisionthe squareoftherelativevelocityisunchanged fromthevalueithadimmediatelybeforethe collision .Yetinitiallythechargeswere comingtogetherandlatertheywereseparat- ing .Therefore,therelativevelocitybefore collisionisdifferentfromthataftercollision, butthesquareisthesame .Itfollowsthat,for reasonsconnectedwithelectromagnetic energyconservation,therelativevelocity

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Anti-gravity electronics

A nti-gravitational behaviour of force-precessed gyroscopes poses perplexingproblems for scientists who are well

versed in the principles of Newtonianmechanics. These machines demonstrate alift force that has no counterpart, as requiredby Newton's Third Law of Motion, unless oneimagines that a balancing force is exerted onthe ether .

Traditionally we have come to accept thataction must he balanced by an equal andopposite reaction . It is true that materialbodies are caused to move by out-of-balanceforces, but they then accelerate and therereally is a balance force represented byinertial reaction . What has now emerged onthe technological scene is a machine thatcan produce that out-of-balance force with-out accelerating . This means that it can,wholly or partially, offset the force of gravita-tion and so defy the acceleration effects ofgravitation .

Some interest in this subject has beenshown in the Feedback columns of Electro-nics and Wireless World (for example, seeJanuary 1987, March 1987 and August 1988issues). For this reason and the furtherreason that the phenomenon has an explana-tion in terms of electronic effects, thefollowing account has been written forE& WW readers .

NEWTON'S RULE

Students of physics, when confronted withNewton's laws of motion, are led to acceptthat when matter interacts by collision orotherwise (e .g. via electric or gravitationalforces) there is separate conservation oflinear momentum and angular momentum-

Action and reaction are balanced and thishas come to mean that no self-actingmachine can develop a propulsive forcewithout shedding mass in some form- Simi-larly, we have always believed that nomachine having a rotor and a stator candevelop its own interaction to rotate therotor at constant speed without applying abalancing torque on the stator .

Textbooks then argue from this action-reaction law and the law of energy conserva-tion that when two perfectly elastic bodiescollide so as to suffer no energy loss by heator radiation they must comply with what isknown as Newton's rule .This rule, you will remember, says that

the relative velocity of the bodies after

Reinterpretation of Newton's Third Law of Motionsuggests that it depends upon an electronic action .

Electronic interaction therefore explains the paradoxicalanti-gravity properties of the force-precessed gyroscope .

H. ASPDEN

THE LAW OF ELECTRODYNAMICS

The Lorentz version of the electrodynamic force law for the force caused by a unit electromagnetic chargemoving at velocity v acting on a unit electromagnetic charge moving at velocity v', separated by a unitvector distance r, can be written as [v' ..[v.r]]in vector product notation. When this same expressionisformulated in scalar product terms it becomes,

(v' .r)v- (v.v') r

This force is not balanced with reaction because it does not lie along rowing to the first term and because itchanges magnitude if v and v' are interchanged .

Clerk Maxwell knew that a terns (v.r)v' could be added without this affecting the empirical data- Such aterm imparts a symmetry which assures balance of linear action and reaction. but allows an out-of-balancecouple .

The alternative- which can be shown to account for induction effects with energy conservation, is tosubtract such a term-

(v' .r)v -- (v,r)v' - (v .v') r

Then the formula assures no out-of-balance couple and so conservation of angular momentum forelectronic interaction- It gives an out-of-balance linear force, but it can satisfy the form of gravitationalinteraction- This is easily seen for the situation in which the gravitational effect is set up by fundamental

charges of the same polarity, same mass and same velocity (v -= v')- Then the first two terms cancel toleave a mutual force of attraction acting directly between the charges and fully satisfying theaction-reaction law of Newton .

impact is - E times the relative velocity ofthe bodies before impact . Here E is what isknown as 'the coefficient of restitution',which has a value of unity for perfectlyelastic loss-free collisions .

What is never explained in textbooks is thechicken-and-egg type of question, namely :'Which comes first, Newton's Third law orNewton's rule?' Why do we take the action-reaction law as fundamental and not New-ton's rule? If Nature actually determines thatNewton's rule is the more fundamental ofthe two, then, given that energy is con-served, we can deduce that action balancesreaction .

Now, of course, it is immaterial to botherabout Nature's priorities if both the action-

reaction law and the rule are unquestionablyvalid in any physical situation . However,having discovered that the action-reactionlaw can be breached, there is purpose inwondering whether Newton's rule is anexpression of a more basic fundamentaltruth .

Then it becomes possible to say that,provided energy associated with the linear,trans .[ationa.[ motion of the interactingbodies is conserved, there will he conserva-tion of linear momentum and so balance ofaction and reaction . However, this argu-ment permits us to imagine that some ofthat energy can be drawn from the rotarymotion of one of the bodies . In this case, wewill not find perfect balance of action andreaction or conservation of linear mornen-

turn . We will, in this special situation, heable to understand how a flywheel can slowdown whilst using its energy to move thesystem linearly against the force of gravita-tion .THE UNDERLYING ELECTRONICS

The implication from this is that Newton'srule is the more fundamental characteristicof interactions between colliding or interact-ing bodies . Flow can electronics be involved?Well, let us not restrict the meaning ofelectronics to the flow of electron currentsin circuits . Electronic action can he that ofthe atomic electrons brought into collisionwith the bodies .

Consider two equal charges of the samepolarity and imagine that they move along acommon line so as to come into collision .Their relative velocity is a measure of themutual electromagnetic field in the nearvicinity of the collision . The energy in thefield at the moment of collision is prop-ortional to that relative velocity squared .Energy is conserved in the collision . There-fore, immediately after the collision thesquare of the relative velocity is unchangedfrom the value it had immediately before thecollision . Yet initially the charges werecoming together and later they were separat-ing. Therefore, the relative velocity beforecollision is different from that after collision,but the square is the same . It follows that, forreasons connected with electromagneticenergy conservation, the relative velocity

after impact is minus one times the relativevelocity before impact . This is Newton's rulefor a loss-free collision .

If the charges have different polarity we donot get the same result, but this merelymeans that the collisions satisfying New-ton's rule are confined to the electronswhich act as the outer guards screening thepositive atomic nuclei from any involve-ment .

The proposition, therefore, is that, whenmatter interacts or collides, the action isreally a summation of actions between fun-damental electron-sized charges. For elec-tromagnetic reasons the action must complywith Newton's rule and this makes that rulethe fundamental condition . Thus the deriva-tion of the law of action and reaction isconsequential upon the requirement that noenergy can transfer from rotary motion tothe linear motion involved in the collision .

It will be seen from this that we have nothad occasion to refer to forces on the ether .We do not need to countenance such forces,because we are not obliged to adhere to theaction-reaction law . However, it is necessaryto find a way in which to force energy fromthe rotary motion of a flywheel, for example .to allow this to be combined with the linearkinetic energy . This is the exceptional role of(he force-precessed gyroscope.

THE FORCE-PRECESSEDGYROSCOPE

It is important to realize that there is noobvious counterpart to Newton's rule whenwe consider rotation . Conservation of angu-lar momentum for motion confined to acommon plane is a direct consequence ofenergy conservation of a body moving underthe action of a central force . When twobodies in rotation collide, the collisionsbetween their individual elementarycharged particle constituents will be thosediscussed in the linear case . However, thereis some fundamental mechanism whichconserves angular momentum and soassures a balance of action and reaction inthat cpnc No doubt this is connected withthat elusive ether or the inertial frame ofreference ; which somehow constitute a uni-versal non-rotating frame of reference .

(b)

Mechanism of force-precessed gyroscope,Middle diagram shows net lift force result-ing from forces F and F' . Lower diagram(the toy gyroscope) shows no anomalousout-of-balance force .

Thus we do find that the gyroscope relieson the principle of conservation of angularmomentum. Imagine that a flywheel spin-ning about a shaft is subjected to a couplewhich acts on the shaft to lend to turn itabout an axis at right angles to the shaft axis .The plane of the flywheel will tend to bedeflected by that couple . This means that theangular momentum can only be conservedabout a given axis if the whole flywheel andits shaft are caused to move about a third axisorthogonal to the two axes already men-tioned .

This motion is that of the precession . In anormal tower-mounted toy gyroscope thegravity forces on the flywheel develop thecouple causing the precession . The centre of

THE FORCE-PRECESSED GYROSCOPE

In the top diagram (above), owing to torque T applied to bearing assembly S' about the vertical axis, thecontra-rotating offset flywheels on pivotally-supported shafts rise, as they precess in a vertical plane .There are no vertical reaction forces on the central support even though the masses of the flywheels arerising. (Gravity forces are disregarded) .

In the middle diagram in order to force the flywheels back to a lower position, forces F are exertedbetween the flywheel shafts and the bearing assembly . This results in complementary forces on thebearing assembly F' acting through the pivots . However, the vertical components of the F' forces, are lessthan the vertical components of the F forces, because the effect of these forces and their reaction is toapply couples to the flywheel shafts which tend to lift the bearing assembly relative to the flywheels . Thismeans that there is an upward thrust P acting on that assembly as it moves in relation to the flywheels fromthe position shown in the top diagram to that shown in the middle diagram . (Again, gravity forces aredisregarded).

In the lower diagram, the effect of relatively weak forces F is depicted, with the precession of theflywheels needed to balance the angular momentum now being about the vertical axis . Here there is no

out-of-balance force. (This disregards gravity forces, but note that such forces due to the weights of theflywheels are analogous in effect to the forces Fin this case .)

Whereas the lower diagram Is representative of the non-anomalous behaviour of the toy gyroscope, acombination of the actions of the top two diagrams can result in a machine with an anomalous net lift force .Such a machine was recently demonstrated by Scott Stratton, an Edinburgh research scientist Hismachine incorporates a cam profile in the bearing surfaces of the bearing assembly . This allows the actionto altercate between a progressive rise of the flywheel shaft and a lift-developing reset by downward thrustimposed via the cam surface . The fact that the machine develops a sustained lift force in defiance ofNewton's Third Law is indisputable, owing to the placement of the demonstration machine on a balancewith a knife edge support and the use of counterweights.

the flywheel processes in a steady horizontalplane around the tower . There is no defianceof any laws of mechanics because the energyof the flywheel is unaffected by the preces-sional motion.

However, imagine now that the couple isnot just that set up by the weight of theflywheel, but is also that of a torque appliedforcibly about the vertical support axis . Theflywheel will then tend to precess in avertical plane and the key question iswhether the energy needed to match thechange of gravitational potential will involveexchange with that of the flywheel rotationor will be drawn from the source poweringthe forced precession . The answer to this .based on observation, is that it is the fly-wheel spin energy that is involved in thegravity balance .

Thp conclusion therefore is that flip

anti-gravitational properties of the force-precessed gyroscope are explicable in termsof the breach of Newton's law of action andreaction, as applied to linear momentumproperties . This has been justified in termsof electronic interaction between matter .

THE FUNDAMENTAL IMPLICATIONIt is curious that it has taken a discoveryconcerning the mechanical properties of thegyroscope to cause us to realize the trueelectronic basis of the laws of mechanics .The evident fact that action need not balancereaction in the linear sense can help toresolve one of the great mysteries in cosmo-logy. Why is it that stars so far removed fromone another can have both linear momen-tum and angular momentum?

If there can be an exchange of energy fromthe spin state to set up linear motion, thenthat need no longer be a problem . Theangular momentum of a star can still bebalanced against that which it possessesowing to its motion around the centre of thegalaxy and the energy exchange can be localto the star .

Of more direct relevance to electronics,however, there is the classical question ofthe electrodynamic interaction between twoelectrons. Anyone who has thought aboutthis will know that the Lorentz force law asused to work out the mutual forces betweentwo electrons in motion gives an out-of-balance linear force and an out-of-balancelinear couple. Physicists excuse this bysaying that all charge motion is circuital andarguing that the out-of-balance effects thencancel out . However, they are wrong in thisand cannot escape the perpetual controversykept alive by those who do believe in thesearch for the real truths .

Ampere is famous for trying to avoid theissue by insisting on a complete balance ofaction and reaction . Maxwell, in his treatise,drew attention to an empirical law whichinsisted on there being no linear out-of-balance but was tolerant of an out-of-balancecouple . I, however, have insisted for thirtyyears that the real truth rests in acceptingthat there has to be no out-of-balancecouple, but there could be an out-of-balancelinear action. This is exactly what hasemerged from the gyroscope experiments .

Why is this important? Well, it comes backto that problem which Einstein could never

solve . I low can the law of electrodynamicsand the law of gravitation be made compati-ble? Remember that Einstein was lockedinto electrodynamics that could he deducedfrom the Lorentz transformations . TheLorentz force law could hardly fit withgravity, which does require a force to actdirectly between the interacting particles .Ampere's old law bore no resemblance togravitation, because it gave different forcesat the same distance for different relativeorientations of the particles and their mo-tion .Equally, the law mentioned by Maxwell

was not of much use, because it involved aturning couple as part of the interaction .This leaves my law and this works for gravity .because the imbalance of linear forcevanishes in the special case of mutuallyparallel charge motion and the law then fitsthe form of the gravity force . However, morethan this, the law is merely based on adding aterm to the Lorentz force to account forFaraday's inductive action .

CONCLUSIONSThanks to the development of force-precessed offset gyroscopic machines it isnow established that Newton's law of actionand reaction balance stands disproved . Thismakes it essential to regard Newton's rule asmore fundamental than his Third Law ofMotion. Newton's rule can he deduced fromelectromagnetic energy conservation asmatter, which is electronic in content, in-

scope life test exceeds 12 yearsOf .1 -

. . .pafter ru nning almost continuouslyfor' '12.-years, twelve . gyroscopesbased oil this design have outlived

their running equipment. Originally, theirCe test was intended to run for two yearsjog :Ministry of Defence funding; tiletract was extended to five dears andi continued ,urh funding from' the

ropeama 'Space' Trrhnolnl y ,Centre andtish':1c!rospace in support of the,Eur--n Space Agency's Olympus cn! •'iication sateihte .lie ),racticahiiit' of expensive con-

unications satei ; :tes depends on . theslong-tern teLahi[ity . of :~'roscope-based

~?rtrtt.itude , contrbi and pointing --systems,This T r!!ru'ar ,)scope the 125 isproduced by .Fe?'raiai for tras, Exosat andthe F:urohean pacciab Instrument Point-ing System . . It is 11s, used in the inertial,guidancc and night cconti'oi s stem forthe

Sane

1

l .a ,nchel,tart from four, for',rOutinc tests;-the

ehve : . ; i rn,rr.>lar" I .r, e 'lily nccasionallyexri ;rdccl bieai .< ,iur

,

r"M.-!

-twelver rnarathon,=;fii :the late •i 970's-iddust-

ral action bk'.,power'ivorker< re cited inlow mains c'o(f Q4 .whit !t ; ; ,J . , ut the

~lufininfi't{

ldr the-that month,tir,e-I' 'men's'stnk~,reduced tl'e mining-time to'

ford-ho ui'ti'a wech t "l. safety reasons . Thegy oscnpes Rave one achieved over tenyears contilluoUs running since the, .sttir`tof the test , andthere t ,) run themfor

n lei 0

January 1989 ELECTRONICS &WIRELESS WORLD

teracts or collides . Starting with Newton'srule and allowing energy conservation todraw on the spin energy of a flywheel there isphysical basis for understanding why anout-of-balance linear force can he produced .

In its t urn . as a pplied . t o the electrodyna-mic charge interaction, this conditionallows the unique law of electrodynamics tohe determined empirically .'I'his law happensto he of the form required to comply withgravitation, hence advancing us towardsthat ultimate goal of field unification . Anincidental result of this is that the differencebetween the Lorentz force law and thatdeduced in this way is precisely that neededin electronic interaction to account toy tileeffects of magnetic induction .

In writing this article no specific referencehas been made to those who deserve praisefor their efforts to get the world to wake up tothe practical significance of the precessinggyroscope's anomalous-force producingproperties .

Supporters of Einstein's theory acclaimEinstein for having shown Ihat Newton's lawof gravitation was inadequate, but arc all tooready to assume that error is involved and soscorn those who demonstrate precessinggyroscopes operating in a way which defiesNewton's laws .

Who then are the pioneers that attract thisattention? Are they just those who havereceived media publicity? So far as the writeris aware, the primary credit of long standinggoes to Alex Jones, Sandy Kidd and EricLaithwaite . but names such as Scott

Strachan and Frederick Scovell are alsolikely to feature in the technological racethat lies ahead . The patent literature extendsbeyond UK and already adds other names ofinventors contributing to this field . This isrevealed by a study commissioned by L .F .Holihan . Director of the Advanced EnergyResearch Institute in London .We are on the verge of a transition

concerning the viability of Newton's ThirdLaw but, since the history of science andinvention cannot he written as it happens,we must await events. In this regard, how-ever, and concerning the author's inter-pretation of the phenomena discussedabove, it is appropriate to note that, inaccepting this article, the Consulting Editorhas stated that he is mindful of similar viewsexpressed to him over many years by AlexJones. This article therefore serves essential-ly to reinforce the prior work of others and .hopefully, will further their cause .

Readers who do not remember the photo-graph showing Professor Laithwaite sup-porting a heavy precessing gyroscopic fly-wheel with his little finger and his armpartially extended should refer to Alex Jones'contribution on p. 64 of the January 1987issue of Eel W1t'. Surely Isaac Newton wouldhave burned out many a candle revising hislaws had he been aware of this phenomenon .

Or. Aspden is in the Department ofE'le ctricalEngineering at the University of South-ampton .

31

u. .

forty ye. .3a a . .d rnl litlast twenty:five years or moreanti-theft devlces Were an optiont ~ . . .~- ..~ ~_~ , , . ., cars but

standard 'n Cadillac. They madeuse of u .e fact that any doorwhich ..,.zned operates a lightswitch -and as there are alsolights ..nder the bonnet andunder ~i .e boot lid, that informa-tion is red into a control unitwhich tomes into operationwhen ~'∎e central-locking systemhas been activated by key .

Any "tempt to open a door,oviinc . or boot immediately setsan alarm off which, in my opin-ion, is the obvious type of alarmin that alternately the hornssounds, followed by all fourheadlights flashing . This goes onfo . ~ty minutes, ceasing justbefo. .hebatteries run down .

It is interesting that if anybodyn.ars a car horn being used theyturn round to see what it is,whereas there are so many alarmnoises now made by sirens etc .that nobody takes any notice ofthem .

1 have all the circuit diagrams. .ailable of the General Motorssystem and in my view it wouldbe quite easy for a person withreasonable intelligence to makeuse of all these door switches asdescribed by Mr Stevens.Joshua SiegerPooleDorset

No integers fora +b --c

1 n,,, 3,.-..as to be no magazinein Australia which invites lettersof the range dealt with by theincomparable Wireless World . Iam sure that this will interestother readers .The following, in which all

. . .fiables are integers greaterthan zero, is offered as a proofthat where n>2; there can be nointegers for a'+b"=c" .Where n>1, x"-y" can always

r:, divided into two factors, oneof which will be x-y. When suchexercises have been carried out afew times it becomes apparentthat the process could be con-

FEEDBACK.u1 .u~J ii .Jenlilt,l'y anJ that U11-number of elements, all positive,within the second set of bracketsequals the value of n. For ex-ample,x 11 -y 11 =(x-y)(x10+yu'+xyf'

+x"y+x`y$+ x"y 2+x3y'+x 'y-f +xay6+x°y°+x' y5 )

Of course, x2-y2=(x-y)(x+y)

and x 3 -y3 =(x-y)(x 2+y2+xy)

x4-y4=(x-y)(x -'+y`+xy2+x2y)

xs-y5=(x-y)(x° +y°+xfy+xy3+x'y2 )

adinfinitum .

So, regarding the equationa2+b2=c2 and the suppositiona"+b"=c", c-a must always be afactor of b" (or of c"-a") andc-b must always be a factor of a"(or of c"-b") .

Regarding the supposition, a"and b" can be factorized inaccordance with these rules onlywhen a, b and c values match setsof integers applicable to theequation; for, if a" and b" arerespectively divisible by c-b andc-a then so must be a2 and b2 .Name and address suppliedWestern Australia

Mine not Wien'sMany thanks for publishing myarticles on "Remotely controlledRC oscillator" in October andNovember of 1988 . 1 would likehowever to make the followingcomment .

The original title of the articlewas "Another look at RC oscilla-tors", this was changed by yourstaff to include the wording "Re-motely controlled Wien oscilla-tors" . Although the Wien-bridgecircuit was referred to in thearticle the various forms of theoscillator described were of myown design and not based on theWien bridge circuit .Austyn J, P . WilliamsRaglanGwent

Anti-gravityelectronics

My article on "Electronic Action

and Reaction" in this issue (p .29)was written before the news thatthe lift forces in the Kiddmachine were confirmed . Read-ers may have seen the front pagestory in the Sunday Express of 23October and the following BBCreports- The device has movedfrom the realm of being a scien-tific curiosity and is headed to-wards commercial technologicalapplication . There aretremendous prospects ahead inthe space and aviation fields .

From the layman's point ofview this is not perpetual mo-tion, but a means of 'swinging'through space, like a Tarzan whocan hook the end of a rope to anychosen point in the sky .

Physicists need somethingmore by way of scientific justi-fication and, with this in mind, Ifeel 1 should comment further onthe electronic explanation in myarticle . The 'relative velocity'proposition from which New-ton's rule is derived is reallybetter formalized in Clerk Max-well's treatise by what is termed'electrokinetic energy' . To derivethe more familiar forms of elec-trodynamic law, Maxwell usedFechner's hypothesis . This saysthat an electronic current is real-ly attributable to a counterfiowof charges of opposite polarity . Inmodern scientific parlance thisimplies electron-positron paircreation and annihilation in away that corresponds to currentflow . I emphasize this because Iwell appreciate the problem ofdefining proper frames of refer-ence for electron collisions,especially where electrons col-lide when moving in the samedirection .

The following references tomy prior published work on thistheme will help readers in-terested in this subject .11 . AspdenDepartment of Electrical Eng .University of Southampton

1 "The Maxwell-Fechner hypothesisas an alternative to Einstein'sTheory", Spec . Sc- Tech ., vol .8, 283(1985) .2 "New perspective on the law ofelectrodynamics", Physics Letters,Vol- 1 I I A, 23 (1985)-

Smps waveformdistortion

I am afraid that Dr Pedder in hisarticle 'How to combat waveformdistortion by switch-mode sup-plies' on page 1016 of the Octo-ber 1988 issue, has got it allwrong. I went through the samecalculations and found that theinput currents are muchsmaller .During recharging of the

capacitor mean current is,

i=Cdwhere dV=20V and

10x20 = 10dt= 180 g ms .

which givesi=450 20

10 9 = B.IA .

Mean current over one period istherefore,

8.1x20 =0-9A180

With the 1A switching convertercurrent, it gives a mean inputcurrent of 1 .9A (not 9A) .

Also, the rms value of thecurrent pulse of trapezoidal formIs,y,,.-(3+(112x10 i/.,)x 20

180

=(3+2.45)9 = 0.605A

and not 3 .2A as Dr Pedder stated .A. BouhadjeraBasingstokeHampshire .

• Dr Pedder comments : DrBouhadjera has misunderstoodthe article . Firstly, 9A is meancurrent level duringthe recharg-ing pulse, not the whole cycle .The recharging current isshown, when idealized, in Fig-3 :15A peak, 9A mean during thepulse and JA mean over thewhole cycle. Secondly, the rmsinput current is 3 .2A as stated .Dr Bouhadjera has somehowcalculated an rms level below hismean level - a form factor of lessthan unity -which would he veryuseful-