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where p is the surface-density, a is the radius of the disk, the origin is the centre of the disk and its plane is the plane of xy. Further,

r2 = x2 + y2+z2, z=r cos 0,

and P stands for Pn (cos 0), Legendre's coefficient of order n; also c, is the coefficient of a" in the expansion of (1+r) powers of x, so that


(1 + x)=co + c1X+C2x2 + €3X3 +....

For the future we shall write cos 0=u, for brevity.

At the plane of the disk (z=0) it is easy to see that the

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two values of V1 are continuous, but that is discontinuous;


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a fact which agrees with what we know from the general properties of the potential. But, apparently, at r=a Vo is not equal to V1; and this is the point which I wish to clear up, for, of course, there can be no discontinuity in the potential and its differential coefficients at any point in free space. From the previous results we have at ra,

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Hence, if there is to be no discontinuity, remembering that P1=μ, we must have

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—μ=(co-c1)+(C1−C2)P2+ • • • + (Cn−Cn+1) P2n+


In order to test this, let us expand fu) in terms of Legendre's coefficients, where

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We know that, with certain restrictions on the nature of f(), of the same type as Dirichlet's conditions for Fourier's sories, we can write

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A1 = (2n+1)
1 + + f(μ)Pn(μ)dμ.


Here the necessary conditions for the expansion are satisfied,

and we find

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{i=¡(2r+ 1) [ { " ̊, ( −μ) P, (u)du + S'" (+m)P, (u)dμ],

A‚= (2r+1) S'μ,(μ)dμ, (r even)

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This gives


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S*P2n−1(14) dμ=cn•


But from the definition of the c's it follows that

and thus


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n = 1

A2n = (Cn − Cn+1) + [(2n−1)cn +2(n+1) Cn+1],

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It follows that the expansion

(Co-c1) + (C1-C2) P2+ • • • ÷ (Cn−Cu+1) P2n+ ·


has the value + if μ>0, and the value -μ if μ<0. Consequently Vo=V1 at r=a, by what has been explained before.


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should also vanish, since there is no surface-density on the sphere. We find that its value is

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+ {2nen + (2n+1)cn+1} P2n+...].

Each of these expressions vanishes, according to the value found above for A2.


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Now V1, Vo satisfy the same differential equation of the second order (Laplace's),

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between 0 and π. It follows that V, must be the analytical continuation of V1 beyond the sphere ra; and the discontinuity at ra is only apparent, not real. A similar point occurs in connexion with the magnetic potential of à circular coil, carrying an electric current; the expressions for this are given in the same article of Thomson and Tait. The numerical details are slightly different, but the principle involved is exactly the same as in the above work.

S. John's College, Cambridge.

26th June, 1901.

XXI. Gas Theory &c.

To the Editors of the Philosophical Magazine. GENTLEMEN,


EFERRING to a quotation on page 40 of your July number concerning the difficulty of reconciling the theoretical and actual ratios of specific heats or elasticities in the case of a permanent gas with diatomic molecules, I have long been accustomed to teach (I hope correctly) that any rotational energy possessed by a dumbbell about its longitudinal axis could have no influence on smooth collisions, and accordingly could not be transferred or altered in amount; therefore such rotation ought not to be included in the partition of energy within the meaning of the Maxwell-Boltzmann law when properly stated. The wording of this law should contain the phrase effective degrees of freedom; and of these there are manifestly 5 in a rigid dumbbell, thus giving at once the theoretical and observed ratio 1.4. The question of vibrational energy is different that has to be, and to some extent has been, dealt with in another manner.


As I am writing may I take the opportunity of congratulating Dr. H. A. Wilson on his successfully destructive criticism, in the same July number, of the over emphasized experiments of M. Crémieu, and of the revolutionary deductions too readily promulgated on the strength of them.

I am, Gentlemen,

Your obedient Servant,

University of Birmingham,
July 13, 1901.

XXII. Notices respecting New Books.

Papers on Mechanical and Physical Subjects. By OSBORNE REYNOLDS, F.R.S., Mem. Inst. C.E., LL.D., Professor of Engineering in the Owens College, and Honorary Fellow of Queen's College, Cambridge. Reprinted from various Transactions and Journals. Vol. II. 1881-1900. Cambridge: at the University Press, 1901. Pp. xii+740.

THE task of grappling successfully with the steadily increasing volume of scientific literature is a problem essentially modern, a problem which did not trouble our forefathers in the seclusion Phil. Mag. S. 6. Vol. 9. No. 8. Aug. 1901.


of those peaceful days when a man could sit down to a book quietly, read it carefully, and thoroughly digest its contents. The rush and hurry of modern life have altered all that. The days when knowledge was so limited that a single individual could master the whole of physical science have long gone by; and one of the great difficulties confronting every worker in science nowadays is that of keeping pace with the resist less tide of modern investigation in only one particular department. The onward sweep of discovery and invention imposes a burden on every specialist which, however fascinating his subject may be, is at times very heavy; and every effort to lighten it must be gratefully welcomed. Most of our scientific societies have already done a great deal towards that object, by publishing abstracts of all the more important papers appearing in contemporary journals. Not less important than this is another method of bringing within easy reach of the student the work of our leading men of science-that of publishing their collected papers. It would be difficult to overestimate the saving in time and trouble which this method effects. Formerly, this was a task which was but seldom undertaken by the author himself, and was generally left to the loving care of his friends. It seems to us much more satisfactory that the author should be his own editor, and this practice has, fortunately, become quite common in our time. The names of Kelvin, Stokes, Heaviside, and Tait will occur to everybody as affording instances of this mode of procedure. And to these may now be added that of Osborne Reynolds.

A truly monumental work is this second volume of Papers, which covers the period 1881-1900 of the author's scientific activity. Many of the papers mark an epoch in the advance of knowledge. Although dealing with a large variety of subjects, the papers of greatest importance are those which are concerned with the motion of fluids-a subject which Professor Osborne Reynolds has made peculiarly his own. Of these, the most important are the following: "An Experimental Investigation of the circumstances which determine whether the Motion of Water shall be direct or sinuous, and of the Law of Resistance in Parallel Channels" [originally published in 1883 in the Phil. Trans.]. "On the Theory of Lubrication and its Application to Mr. Beauchamp Tower's Experiments, including an Experimental Determination of the Viscosity of Olive Oil" [originally published in 1886, Phil. Trans.]; and "On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion" [originally published in Phil. Trans. for 1895]. Besides these, there are several other papers of minor importance dealing with the same subject. A very complete theory of the steamengine indicator and the various errors to which it is liable to give rise, with an account of experimental investigations, is contained in a joint paper by the author and Dr. Brightmore. Towards the end of the volume, on pp. 631-733, we find a very full account of

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