The The Scientific Papers of James Clerk Maxwell, Vol. I Scientific Papers of James Clerk Maxwell, Vol. I
656
The The Scientific Papers of James Clerk Maxwell, Vol. I Scientific Papers of James Clerk Maxwell, Vol. I
656eBook
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Overview
One of the nineteenth century's most significant papers, "A Dynamical Theory of the Electromagnetic Field," appears here, along with similarly influential expositions of Maxwell's dynamical theory of gases. The author's extensive range of interests is well represented, from his discussions of color blindness and the composition of Saturn's rings to his essays on geometrical optics, ether, and protecting buildings from lightning. His less technical writings are featured as well, including items written for the Encyclopedia Britannica and Nature magazine, book reviews, and popular lectures. Striking in their originality, these papers offer a wealth of stimulating and inspiring reading to modern students of mathematics and physics.
Product Details
| ISBN-13: | 9780486783222 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 11/19/2013 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 656 |
| File size: | 33 MB |
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About the Author
James Clerk Maxwell: In His Own Words — And Others
Dover reprinted Maxwell's Treatise on Electricity and Magnetism in 1954, surely one of the first classics of scientific literature over a thousand pages in length to be given new life and accessibility to students and researchers as a result of the paperback revolution of the 1950s. Matter and Motion followed in 1991 and Theory of Heat in 2001.
Some towering figures in science have to speak for themselves. Such is James Clerk Maxwell (1813–1879), the Scottish physicist and mathematician who formulated the basic equations of classical electromagnetic theory.
In the Author's Own Words:
"We may find illustrations of the highest doctrines of science in games and gymnastics, in traveling by land and by water, in storms of the air and of the sea, and wherever there is matter in motion."
"The 2nd law of thermodynamics has the same degree of truth as the statement that if you throw a tumblerful of water into the sea, you cannot get the same tumblerful of water out again." — James Clerk Maxwell
Critical Acclaim for James Clerk Maxwell:
"From a long view of the history of mankind — seen from, say, ten thousand years from now — there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade." — Richard P. Feynman
"Maxwell's equations have had a greater impact on human history than any ten presidents." — Carl Sagan
Read an Excerpt
THE SCIENTIFIC PAPERS OF JAMES CLERK MAXWELL
By W. D. Niven
Dover Publications, Inc.
Copyright © 2003 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-78322-2
CHAPTER 1
I. On the Description of Oval Curves, and those having a plurality of Foci; with remarks by Professor Forbes. Communicated by Professor Forbes.
Mr Clerk Maxwell ingeniously suggests the extension of the common theory of the foci of the conic sections to curves of a higher degree of complication in the following manner:—
(1) As in the ellipse and hyperbola, any point in the curve has the sum or difference of two lines drawn from two points or foci = a constant quantity, so the author infers, that curves to a certain degree analogous, may be described and determined by the condition that the simple distance from one focus plus a multiple distance from the other, may be = a constant quantity; or more generally, m times the one distance+n times the other = constant.
(2) The author devised a simple mechanical means, by the wrapping of a thread round pins, for producing these curves. See Figs. 1 and 2. He then thought of extending the principle to other curves, whose property should be, that the sum of the simple or multiple distances of any point of the curve from three or more points or foci, should be = a constant quantity; and this, too, he has effected mechanically, by a very simple arrangement of a string of given length passing round three or more fixed pins, and constraining a tracing point, P. See Fig. 3. Farther, the author regards curves of the first kind as constituting a particular class of curves of the second kind, two or more foci coinciding in one, a focus in which two strings meet being considered a double focus; when three strings meet a treble focus, &c.
Professor Forbes observed that the equation to curves of the first class is easily found, having the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
which is that of the curve known under the name of the First Oval of Descartes. Mr Maxwell had already observed that when one of the foci was at an infinite distance (or the thread moved parallel to itself, and was confined in respect of length by the edge of a board), a curve resembling an ellipse was traced; from which property Professor Forbes was led first to infer the identity of the oval with the Cartesian oval, which is well known to have this property. But the simplest analogy of all is that derived from the method of description, r and r' being the radients to any point of the curve from the two foci;
mr + nr' = constant,
which in fact at once expresses on the undulatory theory of light the optical character of the surface in question, namely, that light diverging from one focus F without the medium, shall be correctly convergent at another point f within it; and in this case the ratio n/m expresses the index of refraction of the medium.
If we denote by the power of either focus the number of strings leading to it by Mr Maxwell's construction, and if one of the foci be removed to an infinite distance, if the powers of the two foci be equal the curve is a parabola; if the power of the nearer focus be greater than the other, the curve is an ellipse; if the power of the infinitely distant focus be the greater, the curve is a hyperbola. The first case evidently corresponds to the case of the reflection of parallel rays to a focus, the velocity being unchanged after reflection; the second, to the refraction of parallel rays to a focus in a dense medium (in which light moves slower); the third case to refraction into a rarer medium.
The ovals of Descartes were described in his Geometry, where he has also given a mechanical method of describing one of them†, but only in a particular case, and the method is less simple than Mr Maxwell's. The demonstration of the optical properties was given by Newton in the Principia, Book I., prop. 97, by the law of the sines; and by Huyghens in 1690, on the Theory of Undulations in his Traité de la Lumière. It probably has not been suspected that so easy and elegant a method exists of describing these curves by the use of a thread and pins whenever the powers of the foci are commensurable. For instance, the curve, Fig. 2, drawn with powers 3 and 2 respectively, give the proper form for a refracting surface of a glass, whose index of refraction is 1·50, in order that rays diverging from f may be refracted to F.
As to the higher classes of curves with three or more focal points, we cannot at present invest them with equally clear and curious physical properties, but the method of drawing a curve by so simple a contrivance, which shall satisfy the condition
mr + nr' + pr" + &c. = constant,
is in itself not a little interesting; and if we regard, with Mr Maxwell, the ovals above described, as the limiting case of the others by the coalescence of two or more foci, we have a farther generalization of the same kind as that so highly recommended by Montucla, by which Descartes elucidated the conic sections as particular cases of his oval curves.
CHAPTER 2II. On the Theory of Rolling Curves. Communicated by the Rev. Professor Kelland.
There is an important geometrical problem which proposes to find a curve having a given relation to a series of curves described according to a given law. This is the problem of Trajectories in its general form.
The series of curves is obtained from the general equation to a curve by the variation of its parameters. In the general case, this variation may change the form of the curve, but, in the case which we are about to consider, the curve is changed only in position.
This change of position takes place partly by rotation, and partly by transference through space. The rolling of one curve on another is an example of this compound motion.
As examples of the way in which the new curve may be related to the series of curves, we may take the following:—
1. The new curve may cut the series of curves at a given angle. When this angle becomes zero, the curve is the envelope of the series of curves.
2. It may pass through corresponding points in the series of curves. There are many other relations which may be imagined, but we shall confine our attention to this, partly because it affords the means of tracing various curves, and partly on account of the connection which it has with many geometrical problems.
Therefore the subject of this paper will be the consideration of the relations of three curves, one of which is fixed, while the second rolls upon it and traces the third. The subject of rolling curves is by no means a new one. The first idea of the cycloid is attributed to Aristotle, and involutes and evolutes have been long known.
In the History of the Royal Academy of Sciences for 1704, page 97, there is a memoir entitled "Nouvelle formation des Spirales," by M. Varignon, in which he shews how to construct a polar curve from a curve referred to rectangular co-ordinates by substituting the radius vector for the abscissa, and a circular arc for the ordinate. After each curve, he gives the curve into which it is "unrolled," by which he means the curve which the spiral must be rolled upon in order that its pole may trace a straight line; but as this is not the principal subject of his paper, he does not discuss it very fully.
There is also a memoir by M. de la Hire, in the volume for 1706, Part II., page 489, entitled "Methode generale pour réduire toutes les Lignes courbes à des Houlettes, leur generatrice ou leur base étant donnée telle qu'on voudra."
M. de la Hire treats curves as if they were polygons, and gives geometrical constructions for finding the fixed curve or the rolling curve, the other two being given; but he does not work any examples.
In the volume for 1707, page 79, there is a paper entitled, "Methode generale pour déterminer la nature des Courbes formées par le roulement de toutes sortes de Courbes sur une autre Courbe quelconque." Par M. Nicole.
M. Nicole takes the equations of the three curves referred to rectangular co-ordinates, and finds three general equations to connect them. He takes the tracing-point either at the origin of the co-ordinates of the rolled curve or not. He then shews how these equations may be simplified in several particular cases. These cases are—
(1) When the tracing-point is the origin of the rolled curve.
(2) When the fixed curve is the same as the rolling curve.
(3) When both of these conditions are satisfied.
(4) When the fixed line is straight.
He then says, that if we roll a geometric curve on itself, we obtain a new geometric curve, and that we may thus obtain an infinite number of geometric curves.
The examples which he gives of the application of his method are all taken from the cycloid and epicycloid, except one which relates to a parabola, rolling on itself, and tracing a cissoid with its vertex. The reason of so small a number of examples being worked may be, that it is not easy to eliminate the co-ordinates of the fixed and rolling curves from his equations.
The case in which one curve rolling on another produces a circle is treated of in Willis's Principles of Mechanism. Class C. Rolling Contact.
He employs the same method of finding the one curve from the other which is used here, and he attributes it to Euler (see the Acta Petropolitana, Vol. V.).
Thus, nearly all the simple cases have been treated of by different authors; but the subject is still far from being exhausted, for the equations have been applied to very few curves, and we may easily obtain new and elegant properties from any curve we please.
Almost all the more notable curves may be thus linked together in a great variety of ways, so that there are scarcely two curves, however dissimilar, between which we cannot form a chain of connected curves.
This will appear in the list of examples given at the end of this paper.
Let there be a curve KAS, whose pole is at C.
Let the angle DCA = θ1 and CA = r1 and let
θ1 = φ1(r1).
Let this curve remain fixed to the paper.
Let there be another curve BAT, whose pole is B.
Let the angle MBA = θ2, and BA = r2, and let
θ2 = φ2(r2).
Let this curve roll along the curve KAS without slipping.
Then the pole B will describe a third curve, whose pole is C.
Let the angle DCB = θ2, and CB = r3, and let
θ3 = φ3(r3).
We have here six: unknown quantities θ1θ2θ3r1r2r3; but we have only three equations given to connect them, therefore the other three must be sought for in the enunciation.
But before proceeding to the investigation of these three equations, we must premise that the three curves will be denominated as follows:—
The Fixed Curve, Equation, θ1 = φ1(r1).
The Fixed Curve, Equation, θ2 = φ2(r2).
The Fixed Curve, Equation, θ3 = φ3(r3).
When it is more convenient to make use of equations between rectangular co-ordinates, we shall use the letters x1y1, x2y2, x3y3. We shall always employ the letters s1s2s3 to denote the length of the curve from the pole, p1p2p3 for the perpendiculars from the pole on the tangent, and q1q2q3 for the intercepted part of the tangent.
Between these quantities, we have the following equations:—
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
We come now to consider the three equations of rolling which are involved in the enunciation. Since the second curve rolls upon the first without slipping, the length of the fixed curve at the point of contact is the measure of the length of the rolled curve, therefore we have the following equation to connect the fixed curve and the rolled curve—
s1 = s2.
Now, by combining this equation with the two equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
it is evident that from any of the four quantities θ1r1θ2r2 or x1y1x2y2, we can obtain the other three, therefore we may consider these quantities as known functions of each other.
Since the curve rolls on the fixed curve, they must have a common tangent.
Let PA be this tangent, draw BP, CQ perpendicular to PA, produce CQ, and draw BR perpendicular to it, then we have CA = r1, BA = r2, and CB = r3; CQ = p1, PB = p2, and BN = p3; AQ = q1, AP = q2, and CN = q3.
Also
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Since the first curve is fixed to the paper, we may find the angle θ3.
Thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Thus we have found three independent equations, which, together with the equations of the curves, make up six equations, of which each may be deduced from the others. There is an equation connecting the radii of curvature of the three curves which is sometimes of use.
The angle through which the rolled curve revolves during the description of the element ds3, is equal to the angle of contact of the fixed curve and the rolling curve, or to the sum of their curvatures,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
But the radius of the rolled curve has revolved in the opposite direction through an angle equal to dθ2, therefore the angle between two successive positions of r2 is equal to [ds3/r2] - dθ2. Now this angle is the angle between two successive positions of the normal to the traced curve, therefore, if O be the centre of curvature of the traced curve, it is the angle which ds3 or ds1 subtends at O. Let OA = T, then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
As an example of the use of this equation, we may examine a property of the logarithmic spiral.
(Continues...)
Excerpted from THE SCIENTIFIC PAPERS OF JAMES CLERK MAXWELL by W. D. Niven. Copyright © 2003 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Contents
I. On the Description of Oval Curves and those having a plurality of Foci; with remarks by Professor Forbes,II. On the Theory of Rolling Curves,
III. On the Equilibrium of Elastic Solids,
IV. On the Transformation of Surfaces by Bending,
V. On a particular case of the descent of a heavy body in a resisting medium,
VI. On the Theory of Colours in relation to Colour-Blindness,
VII. Experiments on Colour as perceived by the Eye, with remarks on Colour-Blindness,
VIII. On Faraday's Lines of Force,
IX. Description of a New Form of the Platometer, an Instrument for measuring the areas of Plane Figures drawn on paper,
X. On the elementary theory of Optical Instruments,
XI. On a method of drawing the Theoretical Forms of Faraday's Lines of Force without calculation,
XII. On the unequal sensibility of the Foramen Centrale to Light of different Colours,
XIII. On the Theory of Compound Colours with reference to mixtures of Blue and Yellow Light,
XIV. On an instrument to illustrate Poinsot's Theory of Rotation,
XV. On a Dynamical Top, for exhibiting the phenomena of the motions of a body of invariable form about a fixed point, with some suggestions as to the Earth's motion,
XVI. Account of Experiments on the Perception of Colour,
XVII. On the general laws of Optical Instruments,
XVIII. On Theories of the Constitution of Saturn's Rings,
XIX. On the stability of the motion of Saturn's Rings,
XX. Illustrations of the Dynamical Theory of Gases,
XXI. On the Theory of Compound Colours and the Relations of the Colours of the Spectrum,
XXII. On the Theory of Three Primary Colours,
XXIII. On Physical Lines of Force,
XXIV. On Reciprocal Figures and Diagrams of Forces,
XXV. A Dynamical Theory of the Electromagnetic Field,
XXVI. On the Calcidation of the Equilibrium and Stiffness of Frames,