戻る
 講 演 要 旨


数学史からみたポテンシャル論-グリーン関数を中心に
Ivor Grattan-Guinness博士 英国Middlesex 大学名誉教授

編集・解説 八木江里



この講演は既に出版されている論文、“Why did George Green Write His Essay of 1828 on Electricity and Magnetism?”, American Mathematical Monthly,Vol.102,No.5,
pp.387-396,1995をもとにおこなわれる。G-ギネス博士からこの論文を掲載して講演の資料にする許可をいただくことができた。そこで、以下に内容を簡単に紹介してから、論文のコピーを掲載する。
“なぜGeorge Greenは彼の1828年の電気と磁気に関する著作を書いたか?”
George Green(1793−1841、Nottinghamに近いSneintonに生まれる)の生誕200年は1993に英国で盛大におこなわれた。しかしながら彼が後に注目を浴びるポテンシャル論の著作を1828年に発表したときには、殆ど認められなかった。彼は当時、家業の製粉業に従事しながら、限られた時間で数学を研究していた。18世紀の力学の発展では、注目すべき3つの流れがあった。すなわち、Newton, Laplace, Legendreらの回転楕円体の外部への引力の研究、Clairaut,d’Alembert,Euler,Lagrangeらの等ポテンシャル面の研究、Daniel Bernoulliの流体力学の研究で問題になったポテンシャル的な考え方など。
彼Greenが直接的に影響を受けたと思われる先駈者としては,S.D,Poisson (1781-1840)があり、1812年以後、ポテンシャルについての数学的研究を発表していた。Greenが彼の著作でおこなった独自の展開は、電場においてはっきりとpotential function Vと言う用語を導入したことであつた。

(八木追記 私どものグループが研究しているR.Clausiusとの関係について述べると:クラウジウスは、1852―57年に発表した電気理論の4つの論文で、Greenの関数的取り扱いを初めて用いたことを、彼自身の誇りにしている。物理量を力学のモデルに従って関数的にとり扱うことは、1850年の熱力学についての第一論文からおこなっていたが、電気理論の4論文で、Greenの提唱した電位の取り扱い方の関数表示から重要な影響をうけて、potential function V を用いている。詳しくは、私どもの論文集 A Historical Approach to Entropy ,2002, pp.115―123を参照のこと。)

〔以下に論文のコピーを掲載する〕

The American Mathematical Monthly
Volume 102 Number 5/ MAY 1995

1. HONOUR TO GREEN.
Among the centenaries of mathematicians and scientists celebrated in 1993, perhaps the most remarkable was the bicentenary of the birth of a professional miller and part-time mathematician, one George Green (1793-184D of Sneinton, then near Nottingham. Among other achievements, he was the creator of theorems and functions now named after him which make him a principal contributor to potential theory and to its applications in mechanics and mathematical physics.
During the week corresponding to that of his birth (which occurred on 14 July) various events took place. A three-day conference was held at the University of Nottingham, mainly on the use of his work in modern mathematics and physics. It included a visit to the mill at Sneinton, which had been restored and opened as a science centre in 1985. The next day a stained glass window was dedicated at the Gonville and Caius College, Cambridge, where he was resident from 1833 to 1837 as an extremely mature student, and (only) for the winter of 1839-1840 as a Fellow. Finally, on Friday 16 July a meeting was held at the Royal Society of London on his life and work and the modern importance of the latter. It was followed by a quite exceptional event: the unveiling of a plaque in his memory in the floor of the nave of Westminster Abbey, close to the tomb of Isaac Newton and to the plaques for his first publicist Lord Kelvin, Michael Faraday and Clerk Maxwell.
These events had been preceded by the publication in May of an excellent biography of Green (Cannell 1993)-a daunting task to write, as his life is so obscure (for example, no surviving likeness or portrait has ever been found, and his manuscripts seem to have been destroyed). It is clear, though, that in virtual isolation at Sneinton he taught himself Continental mathematics, and produced first-class research work. It was published in 1828 (his 38th year) as a 72-page Essay on the Mathematical Analysis of Electricity and Magnetism (Green 1828), put out at his own expense with the help of a subscription list. Largely ignored during the author's lifetime, it has since been reprinted no less than seven times and translated into German. How and why was it created?

2. THREE STRANDS IN 18TH-CENTURY MECHANICS.
One of the most pro-found influences exercised by this fugitive work is that it raised both the status of potential theory in mathematics and the quality of the theorems that could be stated in it. Prior to this time three strands of thought in potential theory (as we now understand it) were active, though nbt necessarily with close links between them (Todhunter 1873,vol.1).
The most significant strand was the attraction of spheroids to an external point. Isaac Newton had found various special properties in the Principia, in his synthetic
style: they were extended from the 1770s onwards by P. S. Laplace and A. M. Legendre using analytical methods, especially the Legendre functions and surface and zonal harmonics.
Another line came from Alexis Clairaut on the Continent from the 1730s (with some contributions from Colin MacLaurin in Britain soon afterwards), where properties of equipotential surfaces were studied; this work laid stress on the exact differential of a function of several variables, and assisted in the birth soon afterwards of the full partial differential calculus. With d'Alembert, some aspects of Eu]er's work, and especially J. L. Lagrange, variational mechanics was developed, in which force and velocity potentials were often used in the formation of differential equations.
A third strand grew out of Daniel Bernoulli's Hydrodynamica (1738), where considerations of 'ascensis actualis et potentialis' led to conservation of energy as a basis for (much) mechanics; his notions were to end up in the next century as kinetic and potential energy respectively, although with substantial changes in conception in which potential theory was to p]ay a role.
In addition, an isolated contribution came from Lagrange in 1762. While pondering ways of solving the equations for the propagation of sound in three dimensions, he formed volume integrals of the solution in each co-ordinate direction, integrated them by parts to create surface integrals, and then added up the resulting equations to obtain a simpler differential equation to integrate. A clever but ad hoc manoeuvre, it had little influence even upon its distinguished innovator; but it was closer to the way ahead pursued in the early 19th century than the strands just mentioned.

3. POISSON AND THE APPEARANCE OF DIVERGENCE THEOREMS.
Enter Simeon-Denis Poisson (1781-1840), the leading supporting actor in this drama, student and then professor and graduation examiner at the Ecole Polytechnique, devout follower of Laplace and Lagrange in mathematical methods and physical modelling. Poisson inaugurated mathematical electrostatics (1 shall use the word 'electricity' of the time) in two papers (Poisson 1812, 1814) published by the Paris Academy of Sciences in which he analysed arrangements studied experimentally by C.A.Coulomb 30 years previously; equilibrium on a charged spheroid, and between two spheres. The principal mathematical exercise was to modify Legendre functions and related potential theory to fit the assumptions made about the phenomena (Grattan-Guinness 1990, 496-513).
In a short paper written soon after these two, (Poisson 1813) rectified an important oversight of his masters when he pointed out that the differential equation governing the potential X to a body A relative to an interior point M was not Laplace's equation
ΔX = O (with 'Δ' as the Laplacian operator) but ΔX = -4πρ, (1)
where ρwas the density of material at M. He might have got this Insight from his recent work on electricity; another strong candidate is a paper of that year on the attraction of spheroids by Carl Friedrich Gauss, which contained a result which in vectors reads
∫S ds.r = O or = − 4π, where r = BM (2)

and B is an arbitrary point in A, according as M is outside or inside the surface S of A. Neither man dealt with the case where M is on S, when -2π obtains in (1)2 (Grattan-Guinness 1990, 418-424).
Twelve years later Poisson came to the Academy of Science with another pair of papers, this time analysing magnetism (Poisson 1826a-b). Taking a magnetic body A to be composed to discrete 'magnetic elements' D, he set to zero certain surface integrals over D expressing internal equilibrium, and wrote down volume integrals to state the components of attraction of A to an external point M relative to his imposed rectangular co-ordinate system (x, y, z). The second part of the first paper dealt with a 'simplification of the preceding formulae';integrating these integrals by parts with respect to (say) Z led him to convert the volume integral to an integral over the surface S of A. I write his finding in the form

∫∫∫AHz(x,y,z)dxdydz=∫∫sH(xs,ys,zs)cosn dS (3)

where H was the function expressing the components of magnetic attraction, and n was the angle between the z-axis and the normal at the point (xs, ys, zs) of S. Adding this formula to its brothers for the x- and y-directions gave him the first genera] divergence theorem in mathematics;imitating the notation of (3), it can be written
∫∫∫A[Fx+Gy+Hz](x,y,z)dxdydz
=∫∫s[Fcosl+Gcosm+Hcosn] (xs,ys,zs)dS (4)

He modified it for the case when M was inside A by the manner of his proof of (1)2, and found a new term involving a factor -4π/3.
Poisson knew that his result was not restricted to convex bodies (a sum of integrals of the form (3)2 is required as the z-axis goes in and out of A), nor to magnetism. But he saw it simply as a convenience;triple integrals are replaced by double integrals (Grattan-Guinness 1990, 948-953). This point will be crucial for Green, as we shall see the next section.
In a third paper, published by the Academy in the Memoires (Poisson 1827), he analysed the process of magnetisation in moving bodies. A most complicated analysis used Legendre functions once again; but an important detail was his recollection of his equation (1)2 for interior points, and first presentation of the version with -2πρ for surface points.
Surface integrals were enjoying a springtime in French mathematics at this time. For example, Adrien Marie Ampere had been studying electromagnetism and electrodynamics (his word) since 1820; his analysis made adroit use of both surface and line integrals, the latter arising natural]y in connection with the attraction caused by current-bearing wires (Grattan-Guinness 1990, 941-961). One of his most remarkable results, published in 1826, was to show that Poisson's basic formulae for magnetism could be restated in his own preferred conception, which saw magnetism as a special case of electricity and so replaced Poisson's 'magnetic elements' with a tiny electrical solenoid (his word again).
Another source was Joseph Fourier's pioneering work on heat diffusion, created in the mid 1800s, fully published only in the early 1820s, especially in his book Theorie analytique de la chaleur (1822), and then receiving much attention from the new generation of French mathematicians. In particular, around 1826 Jean Duhamel and Russian visitor Mikhail Ostrogradsky independently sought to justify mathematically Fourier's use of trigonometric series solutions (Grattan-Guinness 1990, 1168-1176). Let f and g be two different special solutions for diffusion in a body A, and consider I:=∫∫∫AfgdX. Integrating by parts through A led to a divergence theorem like Poisson's (4); and applying Fourier's external surface condition showed that in fact I = O. Hence f and g were orthogonal over A, like the sine and cosine functions. We can see that this does not provide the justification sought; more to the point is the use again of surface integrals and a divergence theorem.
Although these integrals were making appearance, their presence in mathematics was still slight. Good evidence is provided by Augustin Louis Cauchy (1789-1857), former pupil of the Ecole Polytechnique (when Poisson was professor) and now professor there himself, inaugurating his revision of the calculus and mathematical analysis by his famous new approach with the theory of limits at the centre, emphasis laid upon continuity of functions. (Poisson and others there protested vigorously.) Above all, the derivative and integral were defined separately, so that the fundamental theorem of calculus became a proper theorem for the first time (Grattan-Guinness 1990, 707-804, including Cauchy's concurrent inauguration of complex-variable analysis). However, he never furnished a definition of either the line or the surface integral, although the required forms of definition would not have been hard to devise; they were too marginal to be worth such attention.
Then Green started thinking.

4. GREEN AND THE PLACE OF SURFACE INTEGRALS.
Possible sources for Green's essay will be appraised in the next section; here its main contents are described. Pages are cited from the printing in the edition of his works (Green 1871).
After various preliminaries, the essay contains two roughly equal parts on electricity and on magnetism, in that order. These latter analyses draw heavily on various largely known integral expressions to state external and internal potentials (the latter maybe learnt from Poisson's (1)2), and Legendre functions to express the potentials in analytical form. He extended various results due to Poisson, and considered some variant situations, such as when the spheres are connected by a wire (Whitrow 1984).
The chief novelties were presented in the 'general preliminary results' stated in the opening. First was the explicit specification of 'the potential function,' as he called it (1828, 9), and now named after him:

It only remains therefore to find a function X' which satisfies the partial differential equation, becomes equal to [a given function] X' when [the point] p is upon the surface A, vanishes when p is at an infinite distance from A, and is besides such that none of its differential co-efficients shall be infinite when the point p is exterior to A.

(1828, 12: note the inadequate specification of X'(∞), and that the prime does not denote differentiation). This formulation anticipated in certain ways the ‘Dirichlet principle', which was to assume such status in potential theory when its author began lecturing on the subject from 1839 in Berlin: a decade earlier he also was in Paris, but working on Fourier's heat theory, Cauchy's analysis, and number theory.
Secondly was Green's type of divergence theorem, expressed entirely within the rectangular co-ordinate system (x, y, z) rather than with surface differentials of Poisson's (4): for two 'continuous functions' U(x, y, z) and X(x, y, z)

‘ ∫dxdydzUδX + ∫dσU(dX/dw) = ∫dxdydzVδU+ ∫dσV(du/dw)’ (5)

(1828,23).I follow his use of‘δ’for the Laplacian operator (an unusual symbol,perhaps required by the limitations of his printer’s font box),‘dσ’for the element of the surface, all integrals stated with only one sign ‘∫' (unlike Poisson's use of multiple integral signs), and round brackets to indicate partial ‘differential co-efficients' (Euler's practice, and name also, both of which Green followed). He modified his result for 'singularities' in U (or X) at points G by adding in terms of the form -4πU(xG, yG, zG) to the appropriate side of the equation (1828, 27), like Poisson's own modification; he may also have known of Poisson's equation (1)2 from its reappearance in (Poisson 1827). I wonder at the import of the continuity imposed upon U and X, and the reference to ‘singularities'; had he also been reading Cauchy on reforming the calculus?
Green had taken up a current research interest in mathematical physics in using volume and surface integrals to analyse electricity and magnetism; and with his insights he surpassed all contemporaries. This theorem (5), while similar in mathematical form to Poisson's (4), was understood at a far deeper level as physics (and also surpassed Gauss's (2) in generality). Whereas Poisson saw only simplification in his integral, Green recognised that the importance of his own theorem lay in relating properties inside bodies to properties on their surfaces and vice versa . He must have realised that theorems of this kind served for multiple integral calculus like the fundamental theorem of the calculus itself; hence the importance of integration by parts.
These insights doubtless led Green further to the novelty of his function
X' in which conditions in a body and on its surface were imposed. Such functions were found for various cases with the help of his theorem; one of them followed it in its symmetrical form (Green 1828, 37-39), and launched what have become known as ‘reciprocity relations'.
One may guess therefore, with some confidence, that Poisson's first two papers on magnetism were the source of inspiration for Green 's research , especially the divergence theorem (4). Up to then Green had doubtless been learning mathematical skills and theories, but he had not found a deep problem: Poisson (unintentionally) provided this, in the form of an unexceptionable but somewhat limited use of Legendre functions to analyse the distribution of magnetism, and especially in a 'simplification' which held much deeper consequences than its author had realised.

5. SOURCES AND INFLUENCES.
While it is possible to guess at Green's original motivation, his training in mathematics remains unknown. Among local figures, headmaster John Toplis (1774/1775-1857) would have been a crucial figure in forming the interests of his former pupil at Nottingham Grammar School: a deplorer of the state of British mathematics in the Philosophical Magazine in 1804, a translator of Lacroix there a year later, and of Book One of Laplace's Mecanique celeste in a book published in Nottingham in 1814. However, in 1819 he returned to his college (Queen's, Cambridge), and was not to be one of the subscribers. The only other likely supporter is Sir Edward Bromhead (1789-1855: so Green's senior by a mere four years), member with Charles Babbage and John Herschel of the Analytical Society at Cambridge in the mid 1810s, and a subscriber to the essay; but his letter of April 1828 acknowledging receipt of his presentation copy (Cannell 1993, 67) shows that he had not been aware of its contents before it arrived.
Green's access to literature is also little understood. In his essay he cited as mathematical sources Laplace's Mecanique celeste, Book 3 (1799) for Legendre functions, Fourier's Theorie analytique de la chaleur, and of course, Poisson's three papers on magnetism and the two on electricity; Boit's Traite de physique (1816) was used for information on Coulomb's experiments. A passing reference (1828, 103) to Lagrange's follower L.F.A.Arbogast shows his familiarity with some of the current French operator techniques. A sentence in his introduction comparing Fourier with Cauchy and Poisson on methods of solving differential equations in hydrodynamics(1828,8) suggests that he had read the paper(Fourier 1818) on precisely this matter,which had been published in a Paris journal (Grattan-Guinness 1990,683-686).
How did Green gain access to these works? While British texts would have been available in the local library, access there to foreign literature is much less certain, even presuming that he had funds available to buy it. The point is particularly perplexing for journals-in particular, the Paris Memoires with its Poisson papers. How did Green know that those papers were published there in the first place? Although presentations to the Academy were reported in Paris journals and sometimes abroad, the news did not circulate very much, and Poisson had not given any warning in earlier papers that research in magnetism was in progress. The best chance was that some summary version was translated into a foreign language such as English-and indeed this did happen to summaries of these two papers, in the Quarterly Journal of Science (Poisson 1824,1825).
Each summary paper began with a virtually verbatim repeat of parts of the opening preamble of the parent paper, and then summarised some later results and features. The accounts concentrated mostly on physical and experimental aspects; mathematical procedures were only mentioned (and three formulae quoted in the second summary), although not in a manner to reveal any major novelties. In particular, the divergence theorem (4) was described only in general terms, and with reference to simplication: 'by means of certain transformations, the triple integrals which they contain are reduced to double integrals, and the equations become much more simple' (Poisson 1824, 327). No reader of the time could have guessed that surface integrals were involved; but Green might have been alerted to watch out for the full versions of the papers.
Regarding timetable, the volume of the Paris Academy Memoires containing these papers appeared right at the end of 1826 (Academy of Sciences 1918, 473). Allowing for the usual delay for ships to deliver copies across the channel, one can guess that the spring of 1827 was in hand before Green read at least Poisson's first paper and had his inspiration. Since his essay was to appear in April 1828, this would have given him a maximum of around a year to carry out the research−not an excessive time, even for a part-time mathematician. His motivation was high, most of the required skills and familiarity with the literature were already available −and above all his ideas were fruitful,so that the fruit would grow freely and quickly.

6. OPTIONS FOR PUBLICATION.
However,in contrast to this splendid piece of research and development, Green's sales and marketing were hopeless. He cannot be blamed for his scientific isolation in Nottingham, but he was somewhat naive in resorting to the traditions of publication by public subscription. For the increase in scientific activity in Britain in recent years, together with advances in printing technology, had raised the chances and opportunities for publication, especially for an author like him with financial means available to assist with the costs of production. He could have tried Deighton's of Cambridge, who were then publishing quite a lot in mathematics (Grattan-Guinness 1985) and in fact stocked the essay when it came out; or maybe Taylor (now Taylor and Francis), regular producers of scientific books. He could have written a paper summarising his findings for their Philosophical magazine, which was widely distributed in the scientific world: although it did not publish mathematics frequently, there were papers from time to time, and indeed there had been an exchange in there in 1826 on another aspect of potential theory (namely, properties of equipotential surfaces) between Poisson and the Scottish born mathematician James lvory (Grattan-Guinness 1990, 1190-1195). In fact, if he had felt it proper so to act he could have sought advice from lvory, the mathematician most conversant with potential theory in Britain at that time. He might also have treated his manuscript as a long paper instead of a short book, and tried to submit it to the Royal Society, or the Cambridge Philosophical Society, or the Royal Society of Edinburgh.
I have no doubt that Green never considered any of these possibilities. His essay went to his 52 supporting subscribers, most of whom could not have read a page of it(Green(H.)1946,45-48); and so it vanished from sight. Very rarely has it appeared even in booksellers' catalogues.

7. ON GREEN'S SECOND PERIOD.
Green's later career was somewhat less unorthodox than previously,in as much as he was resident at Gonville and Caius College Cambridge (Bromhead's alma mater) from 1833 to 1837 and for some months of 1839-1840 as a Fellow. He had a small overlap in residence with someone capable of understanding his work, indeed the first mathematician to cite the essay; but this was the eccentric Robert Murphy (1806-1843), who spoilt a promising career by financial incompetence.l
Green published eight papers (and a supplement to one of them),mostly in the Transactions of the Cambridge Philosophical Society with Bromhead as communicator. However, his marketing skills were again to the fore: he cited his essay only twice (1871,120,192),and on neither occasion did he even give the reader the publication details, never mind a comment to explain its importance.
The other papers fall into two partly related groups, both showing strong French influence in both content and methodology (Burkhardt 1908,ch.13). One group deals with elastic bodies,which could be construed to be physically bending objects, or else the elastic aether(and perhaps with luck, both at once).The task was to study the propagation of longitude and transverse vibrations;Green also tackled the difficult question of behaviour at the interface between different substances. He sought generality by making no stipulations about the constituted properties of the substances. The principal influence seems to have been the non-molecular studies of elasticity made from 1827 onwards by Cauchy,which had been partly inspired by Fresnel's work in waval optics (Whittaker 1951,ch.5).
The other group examined the potentials of fluids, which again might cover sound and water, but also the supposed electric and magnetic fluids. Green made this analogy quite explicit in the title of the first of these papers, when referring to the ‘laws of the equilibrium of fluids analogous to the electric fluid' (1871,117).
For methods Green used both his function and theorem, and some of the special results from his essay. He played a little more with operator methods, and produced some solutions in terms of elliptic integrals (although he seemed to be unaware of the recently introduced elliptic functions). A paper on the motion of waves and canals took a step towards the approximating asymptotic solution method now known as 'WKB' (Schlissel 1977,309-314),although he limited himself to working within the linearising models of his time. He worked with potentials to the inverse nth power; and on one occasion he required of his potential function that it be invariant under infinitely small rotations, a step that Sophus Lie was to bring (independently) to great generality and prominence 60 years later.
Somewhat separate from Green's other papers was one dealing with the motion of the 'simple' pendulum. This was a favourite topic at this time, a typical example of small-effect science; for the pendulum was required to work to a great degree of accuracy for the purposes of geodesy. Laplace and Poisson, and also F.W.Bessel and G.B.Airy, had been among its many earlier students (Wolf 1889-1891).

8. RECOGNITION.
Green's marketing skills increased at least to the extent that he sent some of these papers to Carl Jacobi(Cannell 1993,104;the copies are in private possession),and presumably while at Cambridge he gave copies of his essay to William Hopkins,who passed either two or three copies on to the young William Thomson(1824-1907) in 1845.2 Then, as is famously known,the essay found its first enthusiastic reader,four years after the death of its author in 1841.Thomson introduced the name ‘Green's theorem', and soon came to his‘method of images' as result of reading the analysis in the essay of the effect on the electrical charge in a body at an interior/exterior point of a source at a given exterior/interior point (Green 1828,50-54). He soon arranged for the essay to be reprinted in Crelle's journal,although it did not appear until 1850-1854.3 Later he and P.G.Tait called the Dirichlet principle ‘Green's problem' (Thomson and Tait 1883,arts.499-518). The name ‘Green's function' for functions satisfing conditions like Green's own is due to Bernhard Riemann and Carl Neumann (Burkhardt and Mayer 1900,art.18).
Today, Green's function and his theorem are extolled because of the roles which they continue to play in modern physics and in engineering;but it would be a misunderstanding of history to think that their importance is due to these applications. On the contrary,their rise occurred during the period of classical physics,when there appeared a mountainous production of books and papers on potential theory and its use in mathematical physics (Bacharach 1883);all the applications mentioned above were involved, and in due course new ones such as thermodynamics and meteorology,and also mathematical economics. All major applied mathematicians took part,along with many minor ones, and some pure mathematicians also(in particular Karl Weierstrass,who sabotaged standard methods of manipulation in 1870 with his famous counter-example to the Dirichlet principle using the inverse tangent function).
Not only Green's insights and results were used by his successors;his own work, especially the essay, were made available four times in the last thirty years of the 19th century to an extent surpassing all other literature of his own time. The edition of his works by N.Ferrers appeared in London in 1871,and was reprinted in facsimile in 1903 in(of all places)Paris. The essay itself was also reprinted in facsimile,in 1890 in Berlin,in a series of classic reprints of science;five years later it appeared in an annotated German translation by A.van Oettingen and A.Wangerin,in Wilhelm Ostwald's famous booklet series of editions of major scientific works. Green's successors in the classical phase not only absorbed his contributions into their own heritage; they wanted to read the words of the master himself.
Their modern successors have maintained the tradition;for Ferrers's edition appeared again in 1970,and the Essay itself in 1993,in the university of his home town Nottingham,as part of their bicentennial celebrations of their remarkable citizen.

Notes (from pp17-19)
lOn Green's and Murphy's work see(Cross 1985);the reference to Green's book is in(Murphy 1833,587).The nature of Murphy's misdemeanours has not been clear;my information comes from a letter of perhaps 1835 to Babbage written by Augustus De Morgan, who was Professor in London University,where Murphy was then trying to make a living(British Library,Additional Mss.37189,no.241).
Compare(Cannell 1993,112-113).

20ne of these copies of Green's essay is now kept at Nottingham(Cannell 1993, 105), and another is in Keele University Library;I do not know the location of the third one(if there was one).

3The circumstances of this reprinting of the essay are strange.Firstly,Thomson asked Crelle and not his friend Joseph Liouville, who also edited a journal(still often known after Liouville)and was actively interested in potential theory. Secondly,while Crelle had agreed enthusiastically to the suggestion by 1846 (Green (H.),41-43),he did not reprint it for several years, and then in three parts over five years.

REFERENCES
'MAS' denotes the Memoires of the Paris Academy of Sciences, both under that title after the Restoration of 1816 and its Imperial name as the mathematical and physical class of the Institute of France.
Academy of Sciences 1918. Proces-verbaux des seances de l'Academie des Sciences tenues depuis la fondation [in 1795] jusqu'au mois d'aout,1835, vol.8,Hendaye (Observatoire).
Bacharach,M.1883.Abriss der Geschichte der Potentialtheorie,Whrzburg(Thein).
Burkhardt,H,1908.'Entwicklungen nach oscillirenden Functionen und Integration der Differential-gleichungen der mathematischen Physik',Jahresbericht der Deutschen Mathematiker-Vereinigung,10,pt.2,xii+1804 pp.
Burkhardt,H.and Mayer,F.W.F.1900.'Potentialtheorie',in Encyklopddie der mathematischen mssenschaften,vol.2,pt.A,464-503(article IIA7b).
Cannell,D.M.,1993.George Green...,London (Athlone Press).
Cross,J.J.1985.'Integral theorems in Cambridge mathematical physics,1830-55',in (Harman 1985),112-148.
Fourier.J.B.J.1818.'Note relative aux vibrations des surfaces elastiques...', Bulletin des sciences,par la Soeiiti Philomatique de Paris,129-136.[AlsO in Oeuvres,vol.2,255-265.]
Grattan-Guinness I.1985.'Mathematics and mathematical physics at Cambridge 1815-40...’,in(Harman 1985),84-111.
Grattan-Guinness.I.1990.Convolutions in French Mathematics,1800-1840.From the Calculus and Mechanics to Mathematical Analysis and Mathematical physics.3 vols., Basel (Birkhauser) and Berlin (Deutscher Verlag der Wissenschaften).
Green. G.1828.An Essay on the Mathematical Analysis of Electricity and Magnetism, Nottingham(the author).[Repr.in Journal fur die reine und angewandte Mathematik, 39(1850),75-89,44(1852),356-374,47(1854),161-211;1890,Berlin (Mayer and Muller);1958,Goteborg(Ekeldf);1993,Nottingham(The University).Also in(Green 1871),1-ll5.German trans.:1895.Leipzig(Engels-mann:Ostwald's Klassiker der exakten Wissenschaften,no.61).]
Green.G.1871.Mathematical Papers(ed.N.M.Ferrers).London (Macmillan).[Repr.1903, Paris(Hermann);1970,New York (Chelsea).]
Green,H.1946.'A biography of George Green...',in A.Montagu(ed.),Studies and Essays in...Honor of George Sarton,New York,(Schuman),545-594.
Harman,P.M.1985.(Ed.),Wranglers and Physicists...,Manchester(UP).
Murphy,R.1833. 'On the inverse method of definite integrals,...',Transactions of the Cambridge Philosophical Soclety,4,353-408.
Poisson,S.D.1812.'Memoire sur la distribution de lelectricite a la surface des corps conducteurs',MAS,(1811),pt.l,1-92.
Poisson,S.D.1813.'Remarques sur une lequation qui se presente dans la theorie des attractions des spheroides',Nouveau bulletin des sciences,par la Sociati Philomatique de Paris,3(1812-13),388-392.
Poisson,S.D.1814.'Second memoire sur la distribution de l'electricite a la surface des corps conducteurs',M41S,(1811),pt.2,163-274.
Poisson,S.D.1824.[Translation of a summary of Poisson 1826a],Quarterly journa/ of science,17,317-324.[Original in Annales de chimie et de physique,(2)25(1824), 113-137,221-223.]
Poisson,S.D.1825.[Translation of a summary of Poisson 1826b],Quarterly journal of science,19,122-131.[Original in Annales de chimie et de physique,(2)28 (1825), 5-18.]
Poisson,S.D.1826a.'Memoire sur la theorie du magnetisme'.M4S,5(1821-22), 247-338.
Poisson,S.D.1826b.'Second memoire sur la theorie du magnetisme',MAS,5(1821-22), 488-533.
Poisson.S.D.1827.'Memoire sur la theorie du magnetisme en mouvement',MAS,6(1823), 441-570.
Schlissel,A.I.1977.'The development of asymptotic solutions to ordinary differential equations,1817-1920',Archive for History of Exact Sciences,16, 307-378.
Thomson,W.and Tait,P.G.1883.Treatise on Natura/Philosophy,2 pts.Cambridge(UP).
Todhunter,I.1873.A History of the Mathematical Theories ofAttraction and Figure of the Earth...,2 vols.,London (Macmillan).[Repr.1962,New York(Dover)].
Whitrow,G.J.1984.'George Green(1793-1841):a pioneer of modern mathematical physics and its methodology',Annali dell'Istituto di Storia della Scienza di Firenze,no.2,47-68.
Whittaker,E.T.1951.History of the theories of aether and electricity.The classical theories,London (Nelson).
Wolf,C.1889-1891.(Ed.)Mernoires sur le pendule...,2 pts.,Paris
(Gauthler-Villars).

School of Mathematics and Statistics
Middlesex University
Enfield,Middlesex
EN3 4SF,England


戻る