Josephs - Oliver Heaviside Papers Found at Paignton

621.3.011 The Institution of Electrical EngineersMonograph No. 319

Jan. 1959

THE HEAVISIDE PAPERS FOUND AT PAIGNTON IN 1957By H. J. JOSEPHS, Member.

(The paper was received 2nd May, 1958. It was published as an INSTITUTION MONOGRAPH in January, 1959.)

SUMMARYThe papers unearthed dealt mainly with problems connected with

the flow of electromagnetic energy, and showed that Heaviside hadextended Maxwell's theory so that gravitation could be fitted in withelectromagnetism.

It was also found that Heaviside had evolved a rigorous justificationfor his 'operation' of extracting the square root of the process ofpartial differentiation; for he had discovered the conditions underwhich his differential time operator p lost its original significance andbecame the transform parameter of an infinite integral of the Stieltjestype which obeys all the laws of algebra and analysis.

(1) INTRODUCTIONOliver Heaviside lived in Paignton from 1889 to 1897, and

towards the end of 1957 a collection of his papers was foundunder the floorboards of his room in the house where he lived.This collection consisted of a large number of loose sheetsof paper containing his formulae and notes; it also containedmany annotated galley proofs, marked page proofs, scraps ofold letters, envelopes, postcards, technical publications, etc.Altogether the collection filled three sacks. In spite of thedecayed state of some of the papers, it was a simple matter tosort them and connect them with the corresponding parts ofHeaviside's published work. This showed that most of thecollection could be associated with the publication of Volumes 1and 2 of his 'Electromagnetic Theory'.1

Early in 1891 the publishers of the Electrician proposed toHeaviside that his series of articles2 on 'Electromagnetic Theory'just begun in that journal should be brought out later in bookform. The publishers allowed Heaviside great freedom as towhat he wrote in his serial articles, but they insisted that, afterthe first publication in their journal, not a word could be changedfor the later publication as a book. This restriction was imposedin order to keep the costs of production down (for Heaviside'sbooks were not expected to pay), and also because the type setup for the original articles had to be carefully stored (sometimesfor years) until the corresponding sections of the book wereprinted.

Since this restriction thus prevented Heaviside from revisingthe text, it follows that the annotated galley proofs and manu-scripts found at Paignton are of great interest; for they repre-sent Heaviside's second thoughts and show the way he couldhave revised his text had he had the power to do so. He couldhave described his new additions to the mathematical theory ofthe transfer of energy; also he could have shown how Maxwell'stheory can be extended to include both electromagnetic andgravitational phenomena within a single mathematical framework.

To give, within the space of a few thousand words, a completeand satisfactory report of the significance of the mathematicalwork on the scattered sheets in the sacks of paper Heaviside leftbehind at Paignton is well nigh impossible. The most that canbe done in this Monograph is to outline the features whichbecame significant during a careful examination of hiscalculations.

(2) HEAVISIDE'S DUPLEX EQUATIONSThe pencil annotations on the galley proofs, together with the

analysis on the backs of the page proofs, gave a clear indicationof the progress Heaviside made during the years between thefirst publication of his serial articles in the Electrician and then-later publication in book form. This progress lay in thedevelopment of new and improved methods of applying hisduplex circuital equations to electromagnetic problems and theconsequent extension of Maxwell's theory. But before detailsof these new developments are given, something should be saidabout Heaviside's two circuital equations by way of preface.

While Heaviside was living in Camden Town he wrote aseries of articles3 for the Electrician entitled 'ElectromagneticInduction and its Propagation' which he concluded in 1887, twoyears before he moved to Paignton. Heaviside stated that hisobject in writing this series was to present Maxwell's theory in apractical form and to give an adequate discussion of the flow ofenergy in the electromagnetic field.

A few years before Heaviside started this series, Maxwell,expounding Faraday's ideas, had clearly stated that the flow ofenergy depended on the association of two vectors, namely theelectric force E and the magnetic force H. But in 'A Treatiseon Electricity and Magnetism', Maxwell4 did not develop theanalytical consequences of this energy concept; his pages arefilled with descriptions of early Victorian ideas about the natureof electrical energy; moreover, these ideas were expressed in amaze of symbols representing quaternionic formulations ofscalar and vector potential functions, etc. The result was thatengineers found Maxwell's chapter on the general equations ofthe electromagnetic field practically unreadable.

In his Camden Town series3 Heaviside showed engineers thatthe descriptive equations of the electromagnetic field can easilybe based upon the vectorial formulation of two circuital laws.The first states that the electric current / is the curl of themagnetic force H, while the second states that the magneticcurrent Mis the negative curl of the electric force E\ in symbols,

and

/ = curl H .

M = — curl E . . . (2)

Correspondence on Monographs is invited for consideration with a view topublication.

Mr. Josephs is at the Post Office Research Station.

Eqns. (1) and (2) are what electrical engineers call 'Maxwell'sequations'. These equations, however, are not to be found inany of Maxwell's books or papers: they were introduced byHeaviside to act as the basis of a mathematical model of Max-well's theory; they have therefore been called 'Heaviside's equa-tions' in this Monograph, and the technical reasons for thischange are given in the Appendix.

Heaviside's eqn. (1) is the vectorial representation of the well-known equation defining electric current in terms of magneticforce; eqn. (2) was introduced as the proper companion of thefirst in order to make a complete duplex system suitable forengineers to apply to practical problems. Heaviside showedthat this system, based upon the two measurable variables Eand H, readily expresses Maxwell's concept of the flow of energyin the electromagnetic field. He also pointed out that the systemhas the advantage of being quite independent of the unmeasurable

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JOSEPHS: THE HEAVISIDE PAPERS FOUND AT PAIGNTON IN 1957 71

quaternionic formulations of scalar and vector potential functionsintroduced by Maxwell.4

Many of the pencil calculations on the loose sheets found atPaignton arose from Heaviside's applications of his two equa-tions to engineering problems. These calculations showed thathe always kept the duplex features of electromagnetism in mind:in the first place his analysis was naturally symmetrical, since itwas based upon postulated electric charges and magnetic polesand because the inverse-square law was used to define and cal-culate the relevant quantities; furthermore, he manipulated hisequations in a balanced manner, for he always wrote the electricand magnetic currents in eqns. (1) and (2) as

/ = Z> + Jc and M = B + Mc

Here the electric force E and the magnetic force H are relatedto the two fluxes D and B by

D = kE and B =

where k and \L represent permittivity and permeability respec-tively. Also Jc and Mc represent electric and magnetic conduc-tion-current densities and are related to the two forces by

Jc = oE and Mc = omH

where a and om represent electric and magnetic conductivitiesrespectively. It was only in the final computing stage thatHeaviside took account of the zero value of the magnetic con-duction-current density; in all his algebraic manipulative workhe maintained the symmetry of the electrostatic and magneto-static systems.

In these pencil calculations Heaviside wrote the scalar andvector products of his two vectors E and H as EH and YEHrespectively; but this notation is now obsolete, and in thisMonograph these two products are written in the usual wayas E. H and E x H. The vectors E and H are reckoned perunit length and D and B per unit area. The electric energy,U, per unit volume and the magnetic energy, T, per unit volumeare given by the scalar products

while the power-flow vector, P, per unit area is given by thevector product

P = E X H

An examination of his calculations showed that some of themwere connected with the solution of transient problems in cabletelegraphy or telephony: it also became clear that, no matterwhat problem Heaviside was working on, the question of themechanism of energy transfer was touched upon from one angleor another. In these applications he took the space integrals ofU and T to be \CV2 and $LI2 respectively: for he treated theelectric-energy density \E. D as the elemental part of the totalelectric energy \V. CV, where V is the potential between theplates of a condenser whose charge is CV, while he treated themagnetic-energy density \H. B as the elemental part of the totalmagnetic energy £/. LI, where / is the current in a coil of totalinduction LI. V and / were taken as the line integrals of Eand H respectively.

One piece of work was connected with Heaviside's effort tomake the electric and magnetic energies of a telegraph circuitequal (as they must be for distortionless transmission). In sucha circuit the electric energy tends to be excessive, and his remedywas to increase the magnetic energy by increasing the inductanceor to diminish the electric energy by leakance.

For dealing with the transmission of plane waves in a telegraphcircuit Heaviside wrote his circuital equations as

~bH „ . ( )£ , ~bE ^H— ̂ — = oE + k^- and — =r- = u^~

from which he obtained the partial differential equation knownas the 'equation of telegraphy'. Various solutions of thisdifferential equation were found among the scattered papers inthe Paignton collection; many of these solutions had beenpublished by Heaviside, but one unpublished note of particularinterest was found. This showed that he regarded the para-meters in his duplex equations (1) and (2), not as constantquantities, but as statistical distribution functions; to him theproblem of measuring such a parameter was that of determiningits law of distribution and finding its most probable value.Moreover, he regarded his duplex equations as the basis of atractable model of Maxwell's theory which can be used byengineers to predict the results of electrical experiments yet tobe made.

(3) HEAVISIDE'S UNIFIED FIELD THEORYBetween 1883 and 1885, both Heaviside and Poynting5 did

much to establish the mathematical laws describing the flow ofenergy. During this period they independently covered muchthe same ground, and discovered that the rate of energy transferin an electromagnetic field is given by the vector product of Eand H. In 1884, Poynting's paper containing this theorem waspublished a few months before Heaviside's paper; consequentlythe theorem bears Poynting's name.

Poynting's method of establishing the theorem called for themodification and manipulation of Maxwell's original equations,and some heavy mathematical work was involved. Heaviside'smethod, however, was relatively simple and did not call forquaternionic manipulation; after writing the rate of energy lossper unit volume of the field as

- ~

he used his equations (1) and (2) to obtain

divP = H. curl E - E. curl H

Recognizing the right-hand side as the expansion of div (E x H),he saw at once that the rate-of-energy-flow vector is

P=EXH+G (3)

where G is an arbitrary vector representing a circuital energyflux. In 1884, when he established his theorem, Poynting over-looked the existence of the energy flux G; Heaviside, however,while recognizing it, assumed that it represented useless energywhich could be neglected.

The calculations found at Paignton showed that, within tenyears, Heaviside had altered his views about the uselessness ofG in eqn. (3); for he had found that it could be made to satisfythe mathematical requirements of a flux of gravitational energyand that all the energy absorbed by matter in a unified fieldfounded on eqn. (3) could be supplied by the Poynting flux(E x H) alone. Thus, Heaviside appears to have postulated aunified field [based on eqn. (3) and consistent with his viewsabout the electrical constitution of matter] as a means of extend-ing Maxwell's theory to include gravitational phenomena.

Heaviside's expression for the energy density of his unifiedfield consisted of the sum of a number of distinct terms. Oneterm denoted the rate of increase of an electromagnetic energydensity at a point (in the empty or materially occupied space he


72 JOSEPHS: THE HEAVISIDE PAPERS FOUND AT PAIGNTON IN 1957

was considering); other terms represented the rates of increaseof various localized energy densities, such as heat, chemicalenergy, etc. (which he regarded as being due to the electromag-netic field). His expression also contained terms representingthe rate of doing work of certain ponderomotive forces actingon moving matter per unit volume at the point in question. Inthese terms, p was taken to represent the density of matter ande to denote the intensity of a gravitational force which satisfiesNewton's law; Heaviside treated e as the space variation of apotential which depended upon the distribution of matter in hisunified field. The product pe expressed the moving force on pand has its equivalent in the increase of momentum.

Since the law of inverse squares was involved throughout,Heaviside found that the circuital flux of gravitational energycan be expressed as (pit — ce), where u represents the velocity ofp and c is a constant. Expressing this energy flux as the curlof a vector h, Heaviside wrote

curl h = pu — ce (4)

where the divergence of the vector h is arbitrary and may bemade zero. Heaviside took eqn. (4) as a valid gravitationallaw based upon eqn. (3).

The unpublished notes found at Paignton showed that Heavi-side made considerable use of his vector algebra in the develop-ment of eqns. (3) and (4). A good idea of this vectorial treat-ment can be obtained from Appendix B of the first volume of'Electromagnetic Theory'.1 This Appendix, however, was writ-ten before Heaviside had obtained his generalized energyformulations from eqn. (3), upon which he based his unified field.

Many of Heaviside's energy calculations were rather involvedand difficult to interpret, and some of his elemental energy-flowvectors were such that only their total quantity (obtained byintegrating over all space) was observable. It was interesting tonote that Heaviside occasionally manipulated his energy-flowvectors as operational determinants. For example, he wrote thedivergence of his gravitational energy-density vector as

h . (V X e) - e . (V x h)

and then expressed this in its determinantal form as

hi

Viex

hi

v 2e2

hv3 —

e\ e2 e3

V1V2V3/ii h2 h3

where Vi, V2 and V3 are the components of the Hamiltoniandifferentiator V. After replacing the right-hand determinant byits transformed equivalent

hi

Vi

ei

h2

v2 v3*3

—

V1V2V3ex e2 e3

h h2 h3

he obtained the determinantal form of V . (e x A). From thisresult he obtained the energy-density vector (e x h) expressinghis flux of gravitational energy to which other circuital energyfluxes may be added (see Section 9.3).

Other examples illustrating Heaviside's gravitational calcula-tions could be given. In this connection it is interesting to seethe plate facing p. 13 of 'The Heaviside Centenary Volume',6

which is a reproduction of a photograph of an unpublishedgravitational analysis involving his unified field theory; theanalysis was probably made in 1896 and appeared in one ofHeaviside's notebooks which was acquired by The Institutionin 1927.

Heaviside's unified theory discernible in the papers unearthedat Paignton appears to be a straightforward extension of Max-

well's theory. It is based upon a dichotomy of a generalizedenergy law he obtained from eqn. (3), one part producing equa-tions describing the electromagnetic field, and the other thosedescribing the gravitational field. It is interesting to compareHeaviside's ideas with those embodied in Einstein's theory7 ofthe unification of electromagnetism and gravitation publishedmore than half a century later. Einstein's theory shows that ina Euclidean field the quantities concerned separate into a sym-metrical and an antisymmetrical tensor, one set satisfying equa-tions of the electromagnetic type and the other those of thegravitational type. Furthermore, Einstein's analysis showsclearly that the ideas of curved space are unnecessary for thedevelopment of a unified field theory. Thus it appears thatHeaviside's attempt to correlate gravitation with electromag-netism in 1895 has a lot in common with Einstein's attemptin 1950.

(4) HEAVISIDE'S EXPANSION THEOREMSome of the rough calculations found at Paignton indicated

that Heaviside was trying to extend his expansion theorem toinclude cases containing fractional powers of his time operator/?.Notes were found in which he had written his operationalequation as

/(g) _/(Q) , £ fiffr) v(f , m

A(q) A(0) + ? qAXqr)^^ ' ' 'where q is a function of p and F(t, qr) is the equivalent of theoperational expression qfcq — qr). Here qu q2, qj,. . . representthe roots of A(q) = 0, while A'(qr) is the result of substitutingqr for q in dA{q)fdq.

If q is replaced by the differential operator p (= ^fit) thefunction F(/, pr) becomes equivalent to exp (prt) and the inter-pretation of eqn. (5) is a simple matter. But when Heavisidereplaced q by -\/p his function F(/, pr) became equivalent to

and the interpretation of eqn. (5) bristles with mathematicaldifficulties.

The Paignton papers gave no evidence for supposing thatHeaviside overcame the difficulties embodied in the applicationof eqn. (5) to fractional operators. It is highly probable, how-ever, that he abandoned the idea of using it, because in one of hisnotes he wrote '. . . I don't care much about the expansiontheorem now, it is so tame . . .'.

Heaviside's original expansion theorem [obtained from eqn. (5)by putting q = p] has had a peculiar history. It was forgottenand unused for many years; then it was rediscovered and manyengineers regarded it as the subject of the whole of the opera-tional calculus. Heaviside himself, however, regarded it simplyas one useful theorem in the general subject of electric-circuittheory; for it was the first formula to give easily and directlyboth the transient and steady-state responses of a network with-out calling for the determination of arbitrary integration con-stants from initial conditions.

In order to trace Heaviside's method of obtaining solutionsdirectly in terms of initial conditions, the unpublished analysison some of the loose sheets found at Paignton was examined indetail. These sheets were carefully selected, for it was thestudy of energy subsidence in an n-mesh electromagnetic systemwhich led Heaviside to his expansion theorem. In these casesthe solutions were based upon the set of n differential equations,

where

S EnXa = 0

rs == Ori:^p. ' "7w Cfs



In this set Xs represents the dependent variables and the coeffi-cients ars, brs and crs are mesh parameters.

The classical method of solution available to Heaviside wasto substitute for Xs the form as exp (yt), where as and y areconstants. He required n equations for the as with a consistencycondition which determines the 2n possible values of y. Heavi-side soon found, however, that it was impossible for him toformulate the n equations from the knowledge that, at referencetime t — 0, all the currents in the inductors and all the chargeson the condensers are zero. He also found that in the generalcase it was impossible for him to determine the In possible valuesof y. It was at this point that Heaviside parted with classicalmethods and invented his own calculus.

Heaviside's method was to use p instead of ~b[Z)t and then toconsider the equations

S EfSXs = £ [(arsp2 + brsp)us + arspVs]

s

where us and Vs are the values of Xs and dXjdt when t = 0.He then solved for the Xs by algebra as if p were a number, andobtained a quasi-algebraic solution in the form

where frs(p) is a known polynomial in p. He now had toconvert H(p) into an explicit time function C(0, and this he didby expanding H(p) into partial fractions and using his previouslyestablished results such as

and

from which he

(/>•

obtained

P

P- a )

the

tn

~n\tn-l

desired result

H(/>)=C(0, t>0 (6)

(5) HEAVISIDE'S FRACTIONAL DIFFERENTIATIONIn eqn. (6) H(p) represents the operator Heaviside obtained

from his 'algebrized' differential equations, and its conversioninto the explicit time function C(/) constituted his principalproblem.

In the original differential equations from which H(p) wasderived, pn represents d"/^", and its reciprocal p~" denotescorresponding multiple integrations, while the index n is alwaysintegral. But it sometimes happens, as the result of formalalgebraic manipulation, that non-integral or fractional powersof p arise in the operator H(p). This led Heaviside to seek ameaning to ~b"Cl7)tn when n is fractional. 'There is a universeof mathematics', he wrote, 'lying in between the completedifferentiations and integrations.'

Apparently nobody told Heaviside that 'fractional differentia-tion' was an old subject which had been introduced by Leibnizin 1695 and developed by great mathematicians like Euler,Liouville, Gregory, Kelland and others. But Heaviside did nothave access to these authorities, for the reference facilitiesavailable to a poor man working alone in Paignton over 60years ago were practically nil. Nevertheless, he developed thesubject of fractional differentiation much further in some direc-tions than any of his illustrious predecessors.

Heaviside started his attack on the problem by trying togeneralize the equation

nn(fk\ _ ' *k-n

to fractional values of n. After putting k = 0 in this equationhe invented his inverse factorial function g(—n) and establishedthe theorem

rnrg{n)g{—n) = sin rm

for all values of n. Putting n = 1/2 in these equations heobtained the result

where 1 represents his unit function (zero before, unity after,time t = 0). All the algebraic details are given in the secondvolume of 'Electromagnetic Theory'.1 Here Heaviside statesthat he first discovered the value of y/p experimentally; heillustrated this by showing how its value can be deduced fromthe known solution of a heat-flow problem obtained by classicalmethods. This discussion, however, did not appeal to mathe-maticians, since Heaviside's argument did not conform to thestandards of rigour fashionable at the time.

Some of the calculations found at Paignton showed thatHeaviside also arrived at the value of -\/p by another process:this showed him the conditions under which his time operator plost its original significance and became the transform parameterof a function which obeys all the ordinary mathematical rules.Consequently it follows that Heaviside could have constructedrigorous mathematical proofs of his 'fractional differentiation'theorems had he cared. But to Heaviside the construction ofsuch proofs was merely a way of meeting the whims and fanciesof certain pure mathematicians.

These unpublished calculations started with Heaviside usingthe Bessel expansion theorem to write

[ [ f(x)J0[2V(xy)]J0[2V(yt)]dxdyo Jo

where x and t are two positive real variables with differentiatorsa (=bp)x) and p (==hfbi). After replacing the Bessel functionsby their operational equivalents e~yla and e " ^ , Heavisideintegrated with respect to y and obtained

By expanding his operator in inverse powers of p and thenreplacing p~n by /"/«!, Heaviside obtained the function £~O/CT1,which represents his unit impulse placed at the point x = t.Likewise, by expanding the operator in inverse powers of a andthen replacing o~n by xn\n\, he obtained the function e~Pxp\,which represents his unit impulse placed at the point t = x.Thus the two spotting functions are identical, and since theyhave the same symmetrical generator, Heaviside wrote

. . . (7)

In establishing eqn. (7) Heaviside was really arguing in termsof what are now known as Stieltjes integrals; his work, however,was quite independent of that of Stieltjes.8 In eqn. (7) theintegration is against / and consequently the p which occursmust be treated as a parameter and not as an operator.

Replacing H(/>) by y/p\, the 1 which occurs on both sides ofthe equation can be dropped, and Heaviside wrote

r0 0

sJp = f(t)e~ptpdtJo

where f (t) can be determined in the ordinary way. Heavisidesaw at once that f (t) = l/\/(7rt), because on replacing t by x2 heobtained a probability integral whose value is -\/TTI2-\/P.



These calculations must have shown Heaviside that it ismeaningless to talk about 'fractional differentiation', and s/pshould not be interpreted as the 'operation' of extracting thesquare root of the process of partial differentiation, for it is thetransform parameter of an infinite integral which obeys all thelaws of algebra and analysis.

It must be remembered that Heaviside made these calculationswhen the galley proofs of the second volume of his 'Electro-magnetic Theory' were in his hands; consequently, it was thentoo late for him to make changes. It is possible, however,that he may have thought these changes too trivial to botherabout. For Heaviside, like Newton and Laplace, had thefaculty of being able to discover difficult theorems by some sortof intuition which dispensed with the usual processes of proof,and it may have been the possession of this gift that made himso contemptuous of formal logic.

(6) HEAVISIDE'S INFINITE INTEGRALIf the integrand of the infinite integral (7) is divided by

Heaviside's unit impulse function p\ it reduces to the integrandof the famous integral introduced by Laplace in 1812. Nocalculations or notes were found at Paignton to suggest thatHeaviside had ever transformed eqn. (7) into a Laplace integral.This was to be expected, because Heaviside would have realizedimmediately that such a transformation would be a retrogradestep, for it would have destroyed his integration symmetry, sincehis two fundamental functions

would no longer belong together. Moreover, the operationalform of a constant K would no longer be itself but K/p. Conse-quently, all his normal integration processes would have beendisrupted, and his operator H(p) would have differed from f (0in dimensions and serious confusion would result.

In dealing with eqn. (7) by the ordinary method it can be seenthat the dimensions of/? and / must be made reciprocal, in orderto ensure that both the index pt and pdt are dimensionless. Tomake a dimensional check Heaviside would equate f {t) to HQ?);then he would expand f (/) in a convergent series of ascending(positive) powers of /, while H(/>) would be expanded in a con-vergent series of inverse (negative) powers of p. Since f it) andH(/>) are expansible, there is term-by-term correspondence, andthe dimensions of such terms must be identical, e.g.

oo /2r+l oo ( i\r

— (2r 4- IV D 2 r + 1

The dimensional test was one which Heaviside always applied toresults he obtained by direct integration of eqn. (7); and thetransformation of eqn. (7) into the Laplacian form would havedestroyed the dimensional equivalence of f (/) and H(p).

Another result of the Laplacian transformation is that thespotting features of the impulse function in eqn. (7) are elimi-nated, and this would have prevented Heaviside making hisusual topological survey of the problem. For Heaviside wasa born topologist; and he used his invention of operationalspotting functions to help him to visualize intricate relationsbetween abstract 'objects' in his postulated fields. Consequently,Heaviside's method of dealing with eqn. (7) was completely self-contained and did not call for access to a library of books contain-ing details of mathematical transforms. This self-containedfeature was important to Heaviside, for he worked in isolationand his mathematical library at Paignton was very small.

(7) CONCLUSIONThe papers found at Paignton showed that Heaviside was in

line with the British nineteenth-century mathematical traditionof formal algebraic manipulation; he was not an analyst in themodern sense. He was, however, a topologist, and a great one,in his visual thinking about electromagnetism; his ideas, whenproperly interpreted, form a firm foundation upon whichengineering calculations can be based.

(8) REFERENCES(1) HEAVISIDE, O.: 'Electromagnetic Theory' (The Electrician

Printing and Publishing Co., Ltd.), Vol. 1, 1893; Vol. 2,1899; Vol. 3, 1912.

(2) HEAVISIDE, O.: 'Electromagnetic Theory', Electrician, 1890-1899, 26-42.

(3) HEAVISIDE, O.: 'Electromagnetic Induction and its Propaga-tion', ibid., 1884-1887, 14-20.

(4) MAXWELL, J. C : 'Treatise on Electricity and Magnetism'(Clarendon Press, 1873).

(5) POYNTTNG, J. H.: 'On the Transfer of Energy in the Electro-magnetic Field', Philosophical Transactions of the RoyalSociety, 1884, 175, p. 343.

(6) 'The Heaviside Centenary Volume' (The Institution ofElectrical Engineers, 1950).

(7) EINSTEIN, A.: 'The Meaning of Relativity' (Methuen, 1950),Fourth Edition (with an Appendix on a unified fieldtheory).

(8) STIELTJES, T. J.: 'Oeuvres Completes de Thomas JanStieltjes' (Noordhoff, Groningen, 1918), Vol. 2.

(9) HAMILTON, W. R.: 'Elements of Quarternions' (Longmans,London, 1866).

(10) MURPHY, R.: 'The Theories of Electricity' (Pitt Press,London, 1833).

(11) WHTTTAKER, E.: 'Oliver Heaviside', Bulletin of the CalcuttaMathematical Society, 1928-29, 20, p. 202.

(12) GRASSMAN, H. G.: 'Die Lineale Ausdehungslehre, einneuer Zweig der Mathematik' (Reimer, Berlin, 1844).

(13) MAXWELL, J. C.: 'On the Mathematical Classification ofPhysical Quantities', Proceedings of the London Mathe-matical Society, 1871, 3, p. 224.

(9) APPENDICES

(9.1) Heaviside's EquationsIn his epoch-making 'Treatise on Electricity and Magnetism'4

Maxwell's form of presentation was in harmony with the generalprinciples of quaternions as enunciated by Hamilton.? Thepower of quaternions appealed to Maxwell, for near thebeginning of his 'Treatise' (Article 10, Vol. 1, p. 9) he wrote,'In electrodynamics we have to deal with a number of physicalquantities, the relations of which to each other can be expressedfar more simply by a few expressions of Hamilton's than by theordinary equations'.

Maxwell's compact and powerful quaternionic expression ofthe general equations of the electromagnetic field are given inArticle 619, Vol. 2, p. 258, of his 'Treatise'. This formulationappears to have little in common with the vector equations (1)and (2) discussed in this Monograph. Nevertheless, they havecommon features: for both the quaternionic and vector formula-tions describe the experimental discoveries of Oersted, Ampere,Ohm and Faraday; and both formulations can be used to predictthe results of electromagnetic experiments yet to be made.

Hamilton's algebra of quaternions, unlike Heaviside's algebraof vectors, is not a mere abbreviated mode of expressingCartesian analysis, but is an independent branch of mathe-



matics with its own rules of operation and its own specialtheorems. A quaternion is, in fact, a generalized or hyper-complex number: thus a quaternion

w + ix + j'y + kz

is formed from four real numbers, w, x, y and z, and fournumber units, 1, i, j and k, in the same way that the ordinarycomplex number w + ix might be regarded as being formedfrom two real numbers, w and x, and two number units, 1 and i.The number units 1, i, j and k do for rotations and stretches ofa line-element in space what 1 and i do for the line-element ina plane. But whereas the multiplication of complex numbers iscommutative, that of quaternions is not; consequently themanipulation of quaternions calls for the application of certainspecial theorems.

Heaviside also studied quaternions in Hamilton's 'Elements',but had been repelled by the non-commutative property and thespecial theorems. He felt that Hamilton's calculus was really toohard for busy engineers to learn; it would take them too longto master the tricks. So with these ideas in mind he built hisown system of vector algebra. This system was a very simpleone; it was not a new branch of mathematics, like quaternions,but merely a shorthand form of ordinary Cartesian analysis.The only mathematical equipment required to apply this systemwas the definition of scalar and vector products, the Hamiltonianoperator V, a few transformation formula and the integraltheorems of Green and Stokes. Engineers welcomed Heaviside'svector algebra, but pure mathematicians were not enthusiastic.Prof. P. G. Tait described it as 'a sort of hermaphrodite monster,compounded of the notations of Hamilton and Grassman'.Heaviside, however, was not interested in problems of abnormalphysiology, for he was busy forging a mathematical tool whichwould enable engineers to apply Maxwell's electromagnetictheory to their problems.

It was a simple matter for Heaviside to apply his vectoralgebra to the two great experimental laws of electromagnetism.These laws concern the two circuits, the electric circuit and themagnetic circuit which are always linked through a commonfield. By taking the line integral of the magnetic force roundan elemental closed curve in this field, Heaviside obtained hiseqn. (1). This vector equation does not appear explicitly inMaxwell's 'Treatise'; it is possible, however, to find its equivalentin the form of three Cartesian equations in the second volume.This is only to be expected, since these equations describeAmpere's rule for deriving the magnetic force from the electriccurrent and had been discussed in detail by many writers beforeMaxwell (Murphy).10 Considered alone, Heaviside's eqn. (1) ismerely a concise vectorial description of Ampere's results.

By taking the line integral of the electric force round theclosed curve in the field Heaviside obtained his eqn. (2). Againthis vector equation does not appear explicitly in Maxwell's'Treatise'; but since it describes the experimental law of electro-magnetic induction, it is possible to find its equivalent in theform of sets of Cartesian equations in volume 2. These equa-tions may be written in Heaviside's vector notation as

E = - A + Vcurl A = H

where A represents Maxwell's electromagnetic momentum at apoint and «/r denotes his scalar electric potential. Thus Heavi-side's eqn. (2) replaces the above set of equations and eliminatesMaxwell's A and ifj. Considered alone, however, Heaviside'seqn. (2) is merely a concise description of Faraday's results.

Thus, considered individually eqns. (1) and (2) may be directlyrelated to the experimental discoveries upon which electrical

science is based. However, in this Monograph eqns. (1) and (2)are not considered individually but as a correlated or 'duplex' pairof vector equations. These equations have been discussed bySir Edmund Whittaker,11 who stated:

Maxwell, following Faraday's ideas, had clearly pointed out thatthe electric field at each point depended on two vectors, namely theelectric and magnetic forces at the point, and upon the electricand magnetic displacements they produced. But in Maxwell's'Treatise' the analytical consequences of these principles had notbeen developed in a straightforward and natural manner: his pagesare cumbered with the debris of the older theories, with a mazeof symbols representing electric and magnetic potentials, vectorpotentials, and so forth. I well remember, in 1893, buying formyself a second-hand copy of Maxwell which had been the propertyof a College lecturer on mathematical physics. When I came tothe famous chapter on the 'General Equations of the ElectromagneticField', I found scribbled in his handwriting 'from here on this bookis absolutely unreadable'. The great service which Heaviside nowrendered to science was to clear away this accumulation of rubbish,and base the theory on what he called the 'duplex' equations

curl H = 4irTcurl E= — Bdiv T = 0

(where H is the magnetic force, Y is the electric current, etc.),which modern writers generally call 'Maxwell's equations'—thoughthey are not to be found in Maxwell's 'Treatise', and the modernwriters have in fact copied them from Heaviside.

It will be observed that the Heaviside equations quoted bySir Edmund are unsymmetrical, whereas eqns (1) and (2) dis-cussed in this Monograph are perfectly symmetrical. Mostmodern textbooks discuss electromagnetic theory from the view-point of the unsymmetrical equations; they say that the lack ofsymmetry in the system is due to the absence of free magnetismand magnetic conduction. From the Paignton papers, however,it became clear that Heaviside always maintained the electro-magnetic symmetry of eqns. (1) and (2) in his calculations; heretained the magnetic conductivity, am, not merely for the sakeof computing convenience, but also on account of the singularengineering application in which the electric conductivity ismade to perform the functions of both the real a and the unrealam. Thus eqns. (1) and (2) are a pair of correlated vectorequations upon which Heaviside built a mathematical model;this model, however, does not add anything fundamentally newto Maxwell's theory or contain anything new mathematically.It is an algorithm designed by Heaviside to enable engineers toapply Maxwell's theory to their problems.

Heaviside's Paignton calculations showed how he used hismodel: he would write eqns. (1) and (2) in the form

curl H=(^-0

where

and

-curl E = (TZQ)H

F = Propagation parameter

Zo = Characteristic impedance

a + iooK

where a, K, am and /x are the electric conductivity, permittivity,magnetic conductivity and permeability respectively. Thus hemade his discussion of any medium formally uniform with thatof any network; a, K, om and fju for the medium correspondingto G, C, R and L for the network. This mathematical model



made Maxwell's theory workable from the engineer's point ofview. For the model has the properties of an automatic machine;an engineer can feed one end with his parametric facts (or fancies)and on turning the mathematical handle, he can get results fromthe other end—as many as he wants.

(9.2) Maxwell's EquationsIn connection with Sir Edmund Whittaker's comments on

Heaviside's eqns. (1) and (2) it should be noted that the quater-nionic methods used by Maxwell in Article 619 of his 'Treatise'enable these equations to be reduced to the very simple form

where D denotes the quaternionic differentiator and F and Crepresent Maxwell's electromagnetic bivector and currentquaternion respectively. This simple Maxwellian expressionembodies within itself all the fundamental electromagnetic lawsand is an incomparably more powerful mathematical formula-tion than Heaviside's eqns. (1) and (2).

This formulation indicates that Maxwell's mathematical out-look was broader than Heaviside's. To appreciate how muchbroader it was, we must remember that, before Maxwell hadpublished his 'Treatise', he had mastered Grassman's algebra12

and could think in terms of a space of n dimensions, whereasHeaviside, some 25 years later, was still imprisoned in Euclid'sthree dimensions. Thus Maxwell was in possession of analgebra capable of development by specialization in variousdirections: this theory (which included Hamilton's quaternionsas a special case) showed him how to transform his quaternionsinto matrices. The Grassman matrix interpretative schemewould have enabled Maxwell to determine eigenfunctions anduse analytical techniques which are now common in the tensorcalculus.

There can be but little doubt that Maxwell preferred to usequaternionic rather than vector analysis for dealing with reallydifficult electromagnetic problems. For in 1867 he outlined hisown vector analysis;13 his 'convergence' is the negative of the'divergence' in use to-day, and he introduced what is now calledthe 'curl' of a vector. But in his 'Treatise' (published in 1873)he did not use his vector algebra. It appears clear, therefore,that he knew of vector methods before he wrote his 'Treatise'and quite deliberately adopted the quaternionic form of analysisas an improvement. Nobody has yet explained why the Maxwellhypercomplex scheme should not be considered an improvement.

(9.3) Maxwell's Electromagnetic TheoryAlthough the ideas embodied in Heaviside's eqns. (1) and (2)

did not constitute an addition to Maxwell's electromagnetictheory or contain anything new mathematically, the same state-ment cannot be made about the ideas embodied in his develop-ment of eqns. (3) and (4). These ideas may be readily examinedfrom the viewpoint of Maxwell's hypercomplex analysis.Expressing Maxwell's fundamental quaternionic equation

in its tensor form, we have

r = — €rmnEn,

En>n = 0Hn „ = 0

where the dots indicate differentiation with respect to time;ermn is the permutation symbol, and the commas denote partialdifferentiation with respect to the second subscripts. Maxwellshowed that the electric and magnetic forces, Er and Hr, inthese equations are satisfied by

Er = - <f>r -Hr = €rpq<i>q,p

where <j>r (the vector potential) and </r (the scalar potential) areboth functions of position and time. Heaviside found by hisown methods that these functions can be correlated with G inhis eqn. (3) for both Er and Hr are unchanged if <f>r and ift arereplaced by

ft = fa + v, riff' = ijj — V

where v is an arbitrary function of space and time which canbe used to satisfy imposed conditions. If Heaviside could havepublished these Paignton developments (and shown that he couldsatisfy the mathematical requirements of a flux of gravitationalenergy), it is probable that his unified field theory would havebeen accepted as a valid extension of Maxwell's theory.

The above example illustrates the ease with which hyper-complex algebra can be applied to the development of Heavi-side's unified-field ideas. These ideas call for the descriptionof movements in curved Riemannian 4-space, and it is here thatMaxwell's quaternionic formulations are particularly convenient;for movements in such a space can easily be effected by means ofa pair of quaternions, one of which is used as a prefactor and theother as a postfactor. Again Maxwell's quaternionic equationDF = C describing the electromagnetic system S(x, y, z, i) passesreadily into the form D'F' = C" for the system S'(x', y', z', t'),and shows that the invariants of the field are characterized byF2 = 0. Since the electric and magnetic fields can be inter-changed in Maxwell's quaternion by a change of axes, it followsthat the topological characteristics of the two fields can bedetermined.

The metric form for Heaviside's space is

ids)2 = aa - (dx4)2, x4 = ct

For operating in this space it is best to transform Heaviside'seqns. (1) and (2) into the form

T -A- T A- T = 0 * p>n>iT i = 0x rm,n < •*• mn,r ' -* nr,m v> 6 x rm\n v

•HocP = Ta$\ Ea = Ta4 — — Tfa\ 744 = 0where the comma in the suffix indicates partial differentiation,and the vertical stroke denotes covariant differentiation withrespect to n. In this formulation only the fusion of the electricand magnetic fields in the tensor has physical significance. Itwould have been an extremely difficult matter for Heavisideto manoeuvre in this field without the aid of hypercomplexalgebra.


Josephs - Oliver Heaviside Papers Found at Paignton

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