Lord Kelvin Part 3

Further, if q be distributed over an element dS of a surface, the inverse charge aq?f will be distributed over the corresponding element dS' of the inverse surface. But dS'?dS = a4?f4 = f'4?a4 where f, f'

are the distances of O from dS and dS'. Thus if s be the density on dS and s' the inverse density on dS' we have s'?s = a?f' = f?a.

When V is constant over the direct surface, while r has different values for different directions of OQ, the different points of the inverse surface may be brought to zero potential by placing at O a charge -aV. For this will produce at Q' a potential -aV?r' which with V' will give at Q' a potential zero. This shows that V' is the potential of the induced distribution on S' due to a charge -aV at O, or that -V' is the potential due to the induced charge on S' produced by the charge aV at O.

Thus we have the conclusion that by the process of inversion we get from a distribution in equilibrium, on a conductor of any form, an induced distribution on the inverse surface supposed insulated and conducting; and conversely we obtain from a given induced distribution on an insulated conducting surface, a natural equilibrium distribution on the inverse surface. In each case the inducing charge is situated at the centre of inversion. The charges on the conductor (or conductors) after inversion are always obtainable at once from the fact that they are the inverses of the charges on the conductor (or conductors) in the direct case, and the surface-densities or volume-densities can be found from the relations stated above.

[Ill.u.s.tration: FIG. 7.]

Now take the case of two concentric spheres insulated and influenced by a point-charge q placed at a point P between them as shown in Fig. 7. We have seen at p. 49 how the induced distribution, and the amount of the charge, on each sphere is obtained from the two convergent series of images, one outside the outer sphere, the other inside the inner sphere.

We do not here calculate the density of distribution at any point, as our object is only to explain the method; but the quant.i.ties on the spheres S1 and S2, are respectively -q.OA.PB?(OP.AB), -q.OB.AP?(OP.AB).

It may be noticed that the sum of the induced charges is -q, and that as the radii of the spheres are both made indefinitely great, while the distance AB is kept finite, the ratios OA?OP, OB?OP approximate to unity, and the charges to -q.PB?AB, -q.AP?AB, that is, the charges are inversely as the distances of P from the nearest points of the two surfaces. But when the radii are made indefinitely great we have the case of two infinite plane conducting surfaces with a point-charge between them, which we have described above.

Now let this induced distribution, on the two concentric spheres, be inverted from P as centre of inversion. We obtain two non-intersecting spheres, as in Fig. 5, for the inverse geometrical system, and for the inverse electrical system an equilibrium distribution on these two spheres in presence of one another, and charged with the charges which are the inverses of the induced charges. These maintain the system of two spheres at one potential. From this inversion it is possible to proceed as shown by Maxwell in his _Electricity and Magnetism_, vol. i, -- 173, to the distribution on two spheres at two different potentials; but we have shown above how the problem may be dealt with directly by the method of images.

[Ill.u.s.tration: FIG. 8.]

Again take the case of two parallel infinite planes under the influence of a point-charge between them. This system inverted from P as centre gives the equilibrium distribution on two charged insulated spheres in contact (Fig. 8); for this system is the inverse of the planes and the charges upon them. Another interesting case is that of the "electric kaleidoscope" referred to above. Here the two infinite conducting planes are inclined at an angle 360?n, where n is a whole number, and are therefore bounded in one direction by the straight line which is their intersection. The image points I1, J1, ..., of P placed in the angle between the planes are situated as shown in Fig. 3, and are n - 1 in number. This system inverted from P as centre gives two spherical surfaces which cut one another at the same angle as do the planes. This system is one of electrical equilibrium in free s.p.a.ce, and therefore the problem of the distribution on two intersecting spheres is solved, for the case at least in which the angle of intersection is an aliquot part of 360. When the planes are at right angles the result is that for two perpendicularly intersecting planes, for which Fig. 9 gives a diagram.

[Ill.u.s.tration: FIG. 9.]

But the greatest achievement of the method was the determination of the distribution on a segment of a thin spherical sh.e.l.l with edge in one plane. The solution of this problem was communicated to M. Liouville in the letter of date September 16, 1846, referred to above, but without proof, which Thomson stated he had not time to write out owing to preparation for the commencement of his duties as Professor of Natural Philosophy at Glasgow on November 1, 1846. It was not supplied until December 1868 and January 1869; and in the meantime the problem had not been solved by any other mathematician.

As a starting point for this investigation the distribution on a thin plane circular disk of radius a is required. This can be obtained by considering the disk as a limiting case of an oblate ellipsoid of revolution, charged to potential V, say. If Fig. 10 represent the disk and P the point at which the density is sought, so that CP = r, and CA = a, the density is V?{2pv(a - r)}.

The ratio q?V, of charge to potential, which is called the electrostatic capacity of the conductor, is thus 2a?p, that is a?1.571. It is, as Thomson notes in his paper, very remarkable that the Hon. Henry Cavendish should have found long ago by experiment with the rudest apparatus the electrostatic capacity of a disk to be 1?1.57 of that of a sphere of the same radius.

[Ill.u.s.tration: FIG. 10.]

[Ill.u.s.tration: FIG. 11.]

Now invert this disk distribution with any point Q as centre of inversion, and with radius of inversion a. The geometrical inverse is a segment of a spherical surface which pa.s.ses through Q. The inverse distribution is the induced distribution on a conducting sh.e.l.l uninsulated and coincident with the segment, and under the influence of a charge -aV situated at Q (Fig. 11). Call this conducting sh.e.l.l the "bowl." If the surface-densities at corresponding points on the disk and on the inverse, say points P and P', be s and s', then, as on page 51, s' = sa?QP'. If we put in the value of s given above, that of s' can be put in a form given by Thomson, which it is important to remark is independent of the radius of the spherical surface. This expression is applicable to the other side of the bowl, inasmuch as the densities at near points on opposite sides of the plane disk are equal.

If v, v' be the potentials at any point R of s.p.a.ce, due to the disk and to its image respectively, -v' = av?QR. If then R be coincident with a point P' on the spherical segment we have (since then v = V) V' = aV?QP', which is the potential due to the induced distribution caused by the charge -aV at Q as already stated.

The fact that the value of s' does not involve the radius makes it possible to suppose the radius infinite, in which case we have the solution for a circular disk uninsulated and under the influence of a charge of electricity at a point Q in the same plane but outside the bounding circle.

Now consider the two parts of the spherical surface, the bowl B, and the remainder S of the spherical surface. Q with the charge -aV may be regarded as situated on the latter part of the surface. Any other influencing charges situated on S will give distributions on the bowl to be found as described above, and the resulting induced electrification can be found from these by summation. If S be uniformly electrified to density s, and held so electrified, the inducing distribution will be one given by integration over the whole of S, and the bowl B will be at zero potential under the influence of this electrification of S, just as if B were replaced by a sh.e.l.l of metal connected to the earth by a long fine wire. The densities are equal at infinitely near points on the two sides of B.

Let the bowl be a thin metal sh.e.l.l connected with the earth by a long thin wire and be surrounded by a concentric and complete sh.e.l.l of diameter f greater than that of the spherical surface, and let this sh.e.l.l be rigidly electrified with surface density -s. There will be no force within this sh.e.l.l due to its own electrification, and hence it will produce no change of the distribution in the interior. But the potential within will be -2pfs, for the charge is -pfs, and the capacity of the sh.e.l.l is f. The potential of the bowl will now be zero, and its electrification will just neutralise the potential -2pfs, that is, will be exactly the free electrification required to produce potential 2pfs.

To find this electrification let the value of f be only infinitesimally greater than the diameter of the spherical surface of which B is a part; then the bowl is under the influence (1) of a uniform electrification of density -s infinitely close to its outer surface, and (2) of a uniform electrification of the same density, which may be regarded as upon the surface which has been called S above. It is obvious that by (1) a density s is produced on the outer surface of the bowl, and no other effect; by (2) an equal density at infinitely near points on the opposite sides of the bowl is produced which we have seen how to calculate. Thus the distribution on the bowl freely electrified is completely determined and the density can easily be calculated. The value will be found in Thomson's paper.

Interesting results are obtained by diminis.h.i.+ng S more and more until the sh.e.l.l is a complete sphere with a circular hole in it. Tabulated results for different relative dimensions of S will be found in Thomson's paper, "Reprint of Papers," Articles V, XIV, XV. Also the reader will there find full particulars of the mathematical calculations indicated in this chapter, and an extension of the method to the case of an influencing point not on the spherical surface of which the sh.e.l.l forms part. Further developments of the problem have been worked out by other writers, and further information with references will be found in Maxwell's _Electricity and Magnetism_, loc. cit.

It is not quite clear whether Thomson discovered geometrical inversion independently or not: very likely he did. His letter to Liouville of date October 8, 1845, certainly reads as if he claimed the geometrical transformation as well as the application to electricity. Liouville, however, in his Note in which he dwells on the a.n.a.lytical theory of the transformation says, "La transformation dont il s'agit est bien connue, du reste, et des plus simples; c'est celle que M. Thomson lui-meme a jadis employee sous le nom de principe des images." In Thomson and Tail's _Natural Philosophy_, -- 513, the reference to the method is as follows: "Irrespectively of the special electric application, the method of images gives a remarkable kind of transformation which is often useful. It suggests for mere geometry what has been called the transformation by reciprocal radius-rectors, that is to say...." Then Maxwell, in his review of the "Reprint of Papers" (Nature, vol. vii), after referring to the fact that the solution of the problem of the spherical bowl remained undemonstrated from 1846 to 1869, says that the geometrical idea of inversion had probably been discovered and rediscovered repeatedly, but that in his opinion most of these discoveries were later than 1845, the date of Thomson's first paper.[10]

A very general method of finding the potential at any point of a region of s.p.a.ce enclosed by a given boundary was stated by Green in his 'Essay'

for the case in which the potential is known for every point of the boundary. The success of the method depends on finding a certain function, now called Green's function. When this is known the potential at any point is at once obtained by an integration over the surface.

Thomson's method of images amounts to finding for the case of a region bounded by one spherical surface or more the proper value of Green's function. Green's method has been successfully employed in more complicated cases, and is now a powerful method of attack for a large range of problems in other departments of physical mathematics. Thomson only obtained a copy of Green's paper in January 1845, and probably worked out his solutions quite independently of any ideas derived from Green's general theory.

CHAPTER V

THE CHAIR OF NATURAL PHILOSOPHY AT GLASGOW. ESTABLISHMENT OF THE FIRST PHYSICAL LABORATORY

The inc.u.mbent of the Chair of Natural Philosophy in the University of Glasgow, Professor Meikleham, had been in failing health for several years, and from 1842 to 1845 his duties had been discharged by another member of the Thomson gens, Mr. David Thomson, B.A., of Trinity College, Cambridge, afterwards Professor of Natural Philosophy at Aberdeen. Dr.

Meikleham died in May 1846, and the Faculty thereafter proceeded on the invitation of Dr. J. P. Nichol, the Professor of Astronomy, to consider whether in consequence of the great advances of physical science during the preceding quarter of a century it was not urgently necessary to remodel the arrangements for the teaching of natural philosophy in the University. The advance of science had indeed been very great. Oersted and Ampere, Henry and Faraday and Regnault, Gauss and Weber, had made discoveries and introduced quant.i.tative ideas, which had changed the whole aspect of experimental and mathematical physics. The electrical discoveries of the time reacted on the other branches of natural philosophy, and in no small degree on mathematics itself. As a result the progress of that period has continued and has increased in rapidity, until now the acc.u.mulated results, for the most part already united in the grasp of rational theory, have gone far beyond the power of any single man to follow, much less to master.

It is interesting to look into a course of lectures such as were usually delivered in the universities a hundred years ago by the Professor of Natural Philosophy. We find a little discussion of mechanics, hydrostatics and pneumatics, a little heat, and a very little optics.

Electricity and magnetism, which in our day have a literature far exceeding that of the whole of physics only sixty years ago, could hardly be said to exist. The professor of the beginning of the nineteenth century, when Lord Kelvin's predecessor was appointed, apparently found himself quite free to devote a considerable part of each lecture to reflections on the beauties of nature, and to rhetorical flights fitter for the pulpit than for the physics lecture-table.

In the intervening time the form and fas.h.i.+on of scientific lectures has entirely changed, and the change is a testimony to the progress of science. It is visible even in the design of the apparatus. Microscopes, for example, have a perfection and a power undreamed of by our great-grandfathers, and they are supported on stands which lack the ornamentation of that bygone time, but possess stability and convenience. Everything and everybody--even the professor, if that be possible--must be business-like; and each moment of time must be utilised in experiments for demonstration, not for applause, and in brief and cogent statements of theory and fact. To waste time in talk that is not to the point is criminal. But withal there is need of grace of expression and vividness of description, of clearness of exposition, of imagination, even of poetical intuition: but the stern beauty of modern science is only disfigured by the old artificial adornments and irrelevancies.

This is the tone and temper of science at the present day: the task is immense, the time is short. And sixty years since some tinge of the same cast of thought was visible in scientific workers and teachers. The Faculty agreed with Dr. Nichol that there was need to bring physical teaching and equipment into line with the state of science at the time; but they wisely decided to do nothing until they had appointed a Professor of Natural Philosophy who would be able to advise them fully and in detail. They determined, however, to make the appointment subject to such alterations in the arrangements of the department as they might afterwards find desirable.

On September 11, 1846, the Faculty met, and having considered the resolutions which had been proposed by Dr. Nichol, resolved to the effect that the appointment about to be made should not prejudice the right of the Faculty to originate or support, during the inc.u.mbency of the new professor, such changes in the arrangements for conducting instruction in physical science as it might be expedient to adopt, and that this resolution should be communicated to the candidate elected.

The minute then runs: "The Faculty having deliberated on the respective qualifications of the gentlemen who have announced themselves candidates for this chair, and the vote having been taken, it carried unanimously in favour of Mr. William Thomson, B.A., Fellow of St. Peter's College, Cambridge, and formerly a student of this University, who is accordingly declared to be duly elected: and Mr. Thomson being within call appeared in Faculty, and the whole of this minute having been read to him he agreed to the resolution of Faculty above recorded and accepted the office." It was also resolved as follows: "The Faculty hereby prescribe Mr. Thomson an essay on the subject, _De caloris distributione per terrae corpus_, and resolve that his admission be on Tuesday the 13th October, provided that he shall be found qualified by the Meeting and shall have taken the oath and made the subscriptions which are required by law."

At that time, and down to within the last fifteen years, every professor, before his induction to his chair, had to submit a Latin essay on some prescribed subject. This was almost the last relic of the customs of the days when university lectures were delivered in Latin, a practice which appears to have been first broken through by Adam Smith when Professor of Moral Philosophy. Whatever it may have been in the eighteenth century, the Latin essay at the end of the nineteenth was perhaps hardly an infallible criterion of the professor-elect's Latinity, and it was just as well to discard it. But fifty years before, and for long after, cla.s.sical languages bulked largely in the curriculum of every student of the Scottish Universities, and it is undoubtedly the case that most of those who afterwards came to eminence in other departments of learning had in their time acquitted themselves well in the old _Litterae Humaniores_. This was true, as we have seen, of Thomson, and it is unlikely that the form of his inaugural dissertation cost him much more effort than its matter.

[Ill.u.s.tration: PROFESSOR WILLIAM THOMSON, 1846]

The subject chosen had reference no doubt to the papers on the theory of heat which Mr. Thomson had already published. The thesis was presented to the Faculty on the day appointed, and approved, and Mr.

Thomson having produced a certificate of his having taken the oaths to government, and promised to subscribe the formula of the Church of Scotland as required by law, on the first convenient opportunity, "the following oath was then administered to him, which he took and subscribed: _Ego, Gulielmus Thomson, B.A., physicus professor in hac Academia designatus, promitto sancteque polliceor me in munere mihi demandato studiose fideliterque versaturum._" Professor Thomson was then "solemnly admitted and received by all the Members present, and took his seat as a Member of Faculty."

No translation of this essay was ever published, but its substance was contained in various papers which appeared later. The following reference to it is made in an introduction attached to Article XI of his _Mathematical and Physical Papers_ (vol. i, 1882).

"An application to Terrestrial Temperature, of the principle set forth in the first part of this paper relating to the age of thermal distributions, was made the subject of the author's Inaugural Dissertation on the occasion of his induction to the professors.h.i.+p of Natural Philosophy in the University of Glasgow, in October 1846, '_De Motu Caloris per Terrae Corpus_'[11]: which, more fully developed afterwards, gave a very decisive limitation to the possible age of the earth as a habitation for living creatures; and proved the untenability of the enormous claims for TIME which, uncurbed by physical science, geologists and biologists had begun to make and to regard as unchallengeable. See 'Secular Cooling of the Earth, Geological Time,'

and several other Articles below." Some statement of the argument for this limitation will be given later. [See Chap. XIV.]

Thomson thus entered at the age of twenty-five on what was to be his life work as a teacher, investigator, and inventor. For he continued in office fifty-three years, so that the united tenures of his predecessor and himself amounted to only four years less than a century! He took up his duties at the opening of the college session in November, and promptly called the attention of the Faculty to the deficiencies of the equipment of apparatus, which had been allowed to fall behind the times, and required to have added to it many new instruments. A committee was appointed to consider the question and report, and as a result of the representations of this committee a sum of 100 was placed at Professor Thomson's disposal to supply his most pressing needs. In the following years repeated applications for further grants were made and various sums were voted--not amounting to more than 500 or 600 in all--which were apparently regarded as (and no doubt were, considering the times and the funds at the disposal of the Faculty) a liberal provision for the teaching of physical science. A minute of the Faculty, of date Nov.

26, 1847, is interesting.

After "emphatically deprecating" all idea that such large annual expenditure for any one department was to be regularly contemplated, the committee refer in their report to the "inadequate condition of the department in question," and express their satisfaction "with the reasonable manner in which the Professor of Natural Philosophy has on all occasions readily modified his demands in accordance with the economical suggestions of the committee." They conclude by saying that they "view his ardour and anxiety in the prosecution of his profession with the greatest pleasure," and "heartily concur in those antic.i.p.ations of his future celebrity which Monsr. Serville,[12] the French mathematician, has recently thought fit to publish to the scientific world."

Again, in April 1852, the Faculty agree to pay a sum of 137 6_s._ 1_d._ as the price of purchases of philosophical apparatus already made, and approve of a suggestion of the committee that the expenditure on this behalf during the next year should not exceed 50, and "they desire that the purchases shall be made so far as is possible with the previously obtained concurrence of the committee." It is easy to imagine that the ardent young Professor of Natural Philosophy found the leisurely methods of his older colleagues much too slow, and in his enthusiasm antic.i.p.ated consent to his demands by ordering his new instruments without waiting for committees and meetings and reports.

In an address at the opening of the Physical and Chemical Laboratories of the University College of North Wales, on February 2, 1885, Sir William Thomson (as he was then) referred to his early equipment and work as follows: "When I entered upon the professors.h.i.+p of Natural Philosophy at Glasgow, I found apparatus of a very old-fas.h.i.+oned kind.

Much of it was more than a hundred years old, little of it less than fifty years old, and most of it was worm-eaten. Still, with such appliances, year after year, students of natural philosophy had been brought together and taught as well as possible. The principles of dynamics and electricity had been well ill.u.s.trated and well taught, as well taught as lectures and so imperfect apparatus--but apparatus merely of the lecture-ill.u.s.tration kind--could teach. But there was absolutely no provision of any kind for experimental investigation, still less idea, even, for anything like students' practical work. Students'

laboratories for physical science were not then thought of."[13]

It appears that the cla.s.s of Natural Philosophy (there was then as a rule only one cla.s.s in any subject, though supplementary work was done in various ways) met for systematic lectures at 9 a.m., which is the hour still adhered to, and for what was called "Experimental Physics" at 8 p.m.!