Scientific Memoirs/1/On the Forces which regulate the Internal Constitution of Bodies

by Ottaviano-Fabrizio Mossotti

Article XXIII.

On the Forces which regulate the Internal Constitution of Bodies. By O. F. Mossotti.

From a Memoir addressed to M. Plana, published separately, and communicated by M. Faraday, Esq., D.C.L,, F.R.S., &c.

Preliminary Remarks.

1. The study of the phænomena of nature has led philosophers to consider bodies as being composed of molecules held in a state of fixed equilibrium at a certain distance from each other. Such a state requires that they should be endued with a certain action. Some peculiarities of this action we are already able to assign, but its complete characteristics are not yet well defined.

As the resistance opposed by bodies to compression increases indefinitely with the reduction of their volume, though their molecules have not come into contact with each other, it shows that the force which they exercise is repulsive at the least distances. At a distance greater than these, but still imperceptible, it must vary with great rapidity, and become attractive, in order that a steady equilibrium of the molecules may be possible; and finally, when it has become perceptible, it must decrease in the inverse ratio of the square of the distance, in order to represent the universal attraction. The limits of the distance at which the negative action becomes positive vary according to the temperature and nature of the molecules, and determine whether the body which they form be solid, liquid, or aëriform.

There is a class of phænomena, rather singular at first sight, in which however it appears that nature designed, by separating the forces which she employs, to present herself in all her simplicity. Such are the phænomena which constitute what we denominate statical electricity. It is well known with what admirable facility Franklin explained these phænomena, by supposing that the molecules of bodies are surrounded by a quantity of fluid or æther, the atoms of which, while they repel each other, are attracted by the molecules. It is known also how Coulomb subsequently proved that the force with which the repulsion of atoms and the attraction of the molecules are produced, is, like universal attraction, regulated by the law of the inverse ratio of the square of the distance. Indeed, the latter philosopher has substituted for the hypothesis of Franklin, which is that generally followed in England, Germany, and Italy, another hypothesis, in which a second fluid is supposed to perform the part assigned to matter in that of Franklin; and this mode of explaining the phænomena has been more generally adopted in France. It is even asserted that the latter hypothesis is the only one that should be received, inasmuch as it has been completely confirmed by the results of the beautiful analysis with which M. Poisson has begun to enrich the Memoirs of the Academy of Sciences. But they who put forward this assertion have not paid due attention to the fact that, although this illustrious geometer has, for the purpose of establishing his calculations, adopted the language of his school, the inferences drawn from them are not more applicable to the one hypothesis than to the other. He sets out in fact with the principle, that, "If several bodies, being electric conductors, are placed in presence of each other, and attain a permanent state, the result of the actions of the electric layers which cover them, on a point taken anywhere in the interior of a body must, in that state, be null; otherwise the combined electricity which exists in the point under consideration would be decomposed; but this is contrary to the supposed state of permanence." Now if for this principle the following be substituted: "If several bodies, being electric conductors, are placed in presence of each other, and thus attain a permanent state, the result of the actions of the layers of electric fluid which cover them, and of the exterior layers of matter which are not yet neutralized, on the electric fluid at a point taken anywhere in the interior of a body, must, in that state, be null; otherwise the electric fluid which exists in that point would be displaced, which is contrary to the supposed state of permanence;"—and if we interpret accordingly the literal denominations employed by M. Poisson in his equations,—all his results will be equally true on Franklin's hypothesis. In general, the action of the condensed electric fluid will stand for that of the vitreous fluid; and the action exhibited by the matter, in proportion as it is deprived of a quantity of its electric fluid, will stand for that of the resinous fluid. There is one circumstance, however, which makes a difference between the hypothesis of Dufay or Coulomb and that of Franklin: it is this, that, according to the one, the two fluids are moveable in the bodies, while according to the other the electric fluid is, but the matter is not, moveable. As the equilibrium, however, requires that we should only regard the relative position, the mobility of the electric fluid alone is sufficient for its establishment.

Æpinus, who has reduced Franklin's hypothesis to the form of a mathematical theory, was the first to remark, that if it be the requisite condition for the equilibrium of the electric fluids of two bodies, in their natural state, that "the attraction of the matter and the repulsive action of the fluid of the first body on the fluid of the second should be equal, and vice versa," there are but three forces in operation; two of which are attractive, and but one repulsive. In other words, each of the two bodies attracts the fluid of the other, while the mutual repulsion of the two fluids constitutes only a single force, equal to each of the two attractive forces. If then, with the equilibrium of the fluids, it is desired to find the equilibrium of the masses also, an equal repulsion must be allowed between the molecules; since the bodies would otherwise forcibly attract each other. But such an attraction is contrary to what we learn from experience. He felt at first a strong objection to the admission of such a repulsive force between the material molecules, as being opposed to the idea entertained of their mutual attraction, which was so clearly demonstrated on Newton's principles. But a little reflection satisfied him that this admission contained nothing that was opposed to facts, or, as he might rather have said, that was not confirmed by facts. Universal attraction itself may follow as a consequence from the principles which regulate the electric forces: for if we suppose that, the masses being equal, the repulsion of the molecules of matter is a little less than their attraction of the atoms of the æther, or than the mutual repulsion of the atoms themselves, this will be sufficient to leave an excess of attraction which, being directly as the product of the masses and inversely as the square of the distance, would exactly represent the universal attraction.

2. While reflecting on these principles, in a course of lectures on natural philosophy which I gave at the University of Buenos Ayres, I conceived the idea, that if the molecules of matter, surrounded by their atmospheres, attract each other when at a greater, and repel each other when at a less distance, there must be between those two distances an intermediate point at which a molecule would be neither attracted nor repelled, but would remain in steady equilibrium; and that it was very possible this might be the distance at which it would be placed in the the composition of bodies. I thought the idea of sufficient importance to fix it in my memory, but did not at the time pursue its development further.

On my return to Europe I learned, through the reading of some memoirs, and in the course of conversation with men of science, that the attention of geometers was particularly directed to the molecular forces, as being those which may lead us more directly to the knowledge of the intrinsic properties of bodies. I was thus led to recall my ideas on the subject, and set about subjecting them to analysis. The results of my first investigations I here submit to the judgement of philosophers.

I have supposed that a number of material molecules are plunged into a boundless æther, and that these molecules and the atoms of the æther are subject to the actions of the forces required by the theory of Æpinus, and then endeavoured to ascertain the conditions of equililibrium of the æther and the molecules. Considering the æther as a continuous mass, and the molecules as isolated bodies, I found that, if the latter be spherical, they are surrounded by an atmosphere the density of which decreases according to a function of the distance which contains an exponential factor. The differential equation which determines the density being linear, is satisfied by any sum of these functions answering to any number of molecules. Whence it follows that their atmospheres may overlay or penetrate each other without disturbing the equilibrium of the æther. Proceeding in the next place to the conditions of equilibrium of the molecules, I observed that, for a first approximation (which may be sufficient in almost all cases), the reciprocal action of two molecules and of their surrounding atmospheres is independent of the presence of the others, and possesses all the characteristics of molecular action. At first it is repulsive, and contains an exponential factor which is capable of making it decrease very rapidly: it vanishes soon after, and at this distance two molecules would be as much indisposed to approach more nearly as they would be to recede further from each other; so that they would remain in a state of steady equilibrium. At a greater distance the molecules would attract each other, and their attraction would increase with their distance up to a certain point, at which it would attain a maximum: beyond this point it would diminish, and at a sensible distance would decrease directly as the product of their mass, and inversely as the square of their distance.

This action, possessing all the properties with which we can presume that molecular action is endued, is the more remarkable as it has been deduced from those forces only whose existence was already admitted by philosophers, and whose law is characterized by such extraordinary simplicity. When tested in the explanation of the varied phænomena which are proper to it, it must lead, in case of failure, to the exclusion of those forces from amongst physical principles; or, in case of success, establish their reality; and thus mark in a striking manner the admirable œconomy of nature.

To apply the formulæ which we have found, for the purpose of representing molecular action, to the phænomena of the interior constitution of bodies, requires methods of calculation which are not yet developed, and which must become still more complicated when the arrangement of the molecules, their form and their density, are taken into consideration. I have thought it advisable, however, in consideration of tin; use to which it might be applied by able geometers, not to postpone the publication of this mode of viewing molecular action. It is a subject which appears to me entitled to the greatest attention, because the discovery of the laws of molecular action must lead mathematicians to establish molecular mechanism on a single principle, just as the discovery of the law of universal attraction led them to erect on a single basis the most splendid monument of human intellect, the mechanism of the heavens.

Analysis.

3. If several material molecules, which mutually repel each other, are plunged into an elastic fluid, the atoms of which also mutually repel each other, but are at the same time attracted by the material molecules, and if these attractive and repulsive forces are all directly as the masses, and inversely as the square of the distance, it is proposed to determine whether the actions resulting from these forces are sufficient to bring the molecules into equilibrium, and keep them fixed in that state. The object of this inquiry, as may be perceived, is to complete the deductions from the hypothesis of Franklin and Æpinus. It is already known that the conditions of equilibrium which it furnishes, in reference to questions of statical electricity, are in accordance with the phænomena: it remains to be ascertained, whether the molecular actions which result from it are also in accordance with those which determine the interior constitution of bodies. An agreement of this kind would add greatly to the probability that the hypothesis in question is well founded, and afford us a glimpse of the means by which we should be enabled to consider all physical phænomena in one and the same point of view.

Let $f$ be the accelerative force of repulsion existing among the atoms of the æther at a distance taken as unity; $q$ the density at a point $x\,y\,z$, and $\epsilon$ the measure of the elastic force or pressure at the same point, referred to the superficial unit. Let $g$ be the accelerative force of attraction between the atoms of the æther and the matter of the molecules at a distance equal to unity, and ${\bar {\omega }}$ the density at the point $\xi \,\eta \,\zeta$ of a molecule which we suppose to be possessed of a certain though very small extension.

By putting

$F=\iiint {\frac {fq'\,dx'\,dy'\,dz'}{\{(x'-x)^{2}+(y'-y)^{2}+(z-z)^{2}\}^{\frac {1}{2}}}},$

$G=\iiint {\frac {g{\bar {\omega }}\,d\xi \,d\eta \,d\zeta }{\{(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}\}^{\frac {1}{2}}}}$

the triple integral $F$ being extended to the whole space from $x'$, $y'$, $z'$, equal to $-\infty$, as far as $x'$, $y'$, $z'$, equal to $=\infty$ (the small parts occupied by the molecules being excepted), and the triple integral $G$ being extended to all the values of $\xi$, $\eta$, $\zeta$, that answer to the points occupied by the molecule, we shall have for the equilibrium of the æther the equations

${\frac {d\epsilon }{dx}}=-q{\frac {dF}{dx}}+q{\frac {dG}{dx}}+q{\frac {dG_{1}}{dx}}+q{\frac {dG_{2}}{dx}}\ldots \ldots +q{\frac {dG_{\nu }}{dx}}+\mathrm {etc.}$

(I)

${\begin{aligned}&{\frac {d\epsilon }{dy}}=-q{\frac {dF}{dy}}+q{\frac {dG}{dy}}+q{\frac {dG_{1}}{dy}}+q{\frac {dG_{2}}{dy}}\ldots \ldots +q{\frac {dG_{\nu }}{dy}}+\mathrm {etc.} \\&{\frac {d\epsilon }{dz}}=-q{\frac {dF}{dz}}+q{\frac {dG}{dz}}+q{\frac {dG_{1}}{dz}}+q{\frac {dG_{2}}{dz}}\ldots \ldots +q{\frac {dG_{\nu }}{dz}}+\mathrm {etc.} \end{aligned}}$

in which $G_{1}$, $G_{2}$, … $G_{\nu }$, &c. denote the quantities analogous to $G$ which correspond with the different molecules 1, 2 … $\nu$, &c.

Let us likewise put

${\begin{aligned}&\Phi =\iiint {\frac {gq'\,dx'\,dy'\,dz'}{\{x'-xi)^{2}+(y'-\eta )^{2}+(z'-\zeta )^{2}\}^{\frac {1}{2}}}}\\&\Gamma _{\nu }=\iiint {\frac {\gamma {\bar {\omega }}_{\nu }\;d\xi _{\nu }\;d\eta _{\nu }\;d\zeta _{\nu }}{\{(\xi _{\nu }-\xi )^{2}+(\eta _{\nu }-\eta )^{2}+(\zeta _{\nu }-\zeta )^{2}\}^{\frac {1}{2}}}}\end{aligned}}$

where $\gamma$ denotes the force of repulsion existing among the molecules of matter at the distance assumed as unity.

The equations for the equilibrium of a molecule, if we take into consideration the motion of its centre of gravity only, will be

(II)

${\begin{aligned}&\iint \epsilon \,d\eta \,d\zeta =\iiint {\bar {\omega }}{\frac {d\Phi }{d\xi }}\,d\xi \,d\eta \,d\zeta -\Sigma \iiint {\bar {\omega }}{\frac {d\Gamma _{\nu }}{d\xi }}\,d\xi \,d\eta \,d\zeta \\&\iint \epsilon \,d\xi \,d\zeta =\iiint {\bar {\omega }}{\frac {d\Phi }{d\eta }}\,d\xi \,d\eta \,d\zeta -\Sigma \iiint {\bar {\omega }}{\frac {d\Gamma _{\nu }}{d\eta }}\,d\xi \,d\eta \,d\zeta \\&\iint \epsilon \,d\xi \,d\eta =\iiint {\bar {\omega }}{\frac {d\Phi }{d\zeta }}\,d\xi \,d\eta \,d\zeta -\Sigma \iiint {\bar {\omega }}{\frac {d\Gamma _{\nu }}{d\zeta }}\,d\xi \,d\eta \,d\zeta \end{aligned}}$

The sum $\Sigma$ is to be extended to all the numbers $\nu$, that is to say, to all the molecules except that one the equilibrium of which we are considering; the double integral is to be extended to the whole surface of this molecule, and the triple integrals to its whole volume.

4. Let us begin by considering the equilibrium of the æther. The elasticity possessed by the æther at any point of space can be only the result of the reciprocal action of the contiguous parts: hence we are led, by considerations analogous to those employed by Laplace in reference to the repulsion of caloric, in the 12th book of the Mécanique Céleste, to conclude that, in a fluid considered as a continuous mass, the elasticity is proportional to the square of the density. If then $k$ represents a constant coefficient, we shall have $\epsilon ={\frac {1}{2}}kq^{2}$, and by substituting this value in the equations (I) we shall derive the following:

$k{\frac {dq}{dx}}=-{\frac {dF}{dx}}+{\frac {dG}{dx}}+{\frac {dG_{1}}{dx}}+{\frac {dG_{2}}{dx}}\ldots \ldots +{\frac {dG_{\nu }}{dx}}+\mathrm {etc.}$

(I)'

${\begin{aligned}k{\frac {dq}{dy}}&=-{\frac {dF}{dy}}+{\frac {dG}{dy}}+{\frac {dG_{1}}{dy}}+{\frac {dG_{2}}{dy}}\ldots \ldots +{\frac {dG_{\nu }}{dy}}+\mathrm {etc.} \\k{\frac {dq}{dz}}&=-{\frac {dF}{dz}}+{\frac {dG}{dz}}+{\frac {dG_{1}}{dz}}+{\frac {dG_{2}}{dz}}\ldots \ldots +{\frac {dG_{\nu }}{dz}}+\mathrm {etc.} \end{aligned}}$

which lead directly to the complete integral

(III)

$kq=C-F+G+G_{1}+G_{2}\ldots \ldots G_{\nu }+\mathrm {etc.} ;$

$C$ being an arbitrary constant.

In order to determine, by means of this equation, the density $q$, we must substitute for $F$, $G$, $G_{1}$, $G_{2}$, … $G_{\nu }$, &c. the integrals which they represent. If the rectangular co-ordinates are changed into polar co-ordinates by means of the known formulæ

${\begin{aligned}x&=r\sin \theta \cos \psi &y&=r\sin \theta \sin \phi &z&=r\cos \theta \\x'&=r'\sin \theta '\cos \psi '&y'&=r'\sin \theta '\sin \psi '&z'&=r'\cos \theta '\end{aligned}}$

the expression for $F$ takes the form (see the additions to the Connaissance des Temps for the year 1829, p. 356)

(IV)

${\begin{aligned}F&=\Sigma _{0}^{\infty }\left[{\frac {q}{r^{n+1}}}\iint \left(\int _{0}^{r}fq'r^{'n+2}\;dr'\right)P_{n}\sin \theta '\,d\theta '\,d\psi '\right]\\&+\Sigma _{0}^{\infty }\left[r^{n}\iint \left(\int _{r}^{\infty }{\frac {fq'}{r^{'n+1}}}\right)P_{n}\sin \theta '\,d\theta '\,d\psi '\right]\end{aligned}}$

The coefficient $P_{n}$ being given by the formula

$P_{n}={\frac {1.3.5..2n-1)}{1.2.3\dots \dots .n}}$

$\left\{p^{n}-{\frac {n(n-1)}{2(2n-1)}}p^{n-2}+{\frac {n(n-1)(n-2)(n-3)}{2.3(2n-1)(2n-3)}}p^{n-4}+\mathrm {etc.} \right\}$

in which

$p=\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos(\psi -\psi '),$

and the limits of the integrals relative to $\theta '$ and $\psi '$ should be such that the value of $F$ may take in the whole space, except the small portions occupied by the material molecules.

In order to have the expression for $G$, let us in like manner put

$\xi =\rho \sin \omega \cos \phi ,\qquad \eta =\rho \sin \omega \sin \phi ,\qquad \zeta =\rho \cos \omega$

and represent by $\Pi _{n}$ the function $P_{n}$, when $r'$, $\theta '$, $\psi '$, are therein changed into $\rho$, $\omega$, $\phi$. Then, if we suppose the origin of the co-ordinates to be taken in the interior of the molecule, we shall have (see Connaissance des Temps for the year 1829, p. 357)

(V)

$G=\Sigma _{0}^{\infty }\left[{\frac {1}{r^{n+1}}}\iint \left(\int _{0}^{r}g{\bar {\omega }}\rho ^{n+2}\,d\rho \right)\Pi _{n}\sin \omega \,d\omega \,d\phi \right]$

${\begin{aligned}&+\Sigma _{0}^{\infty }\left[{\frac {1}{r^{n+1}}}\iint \left(\int _{0}^{r}g{\bar {\omega }}\rho ^{n}+2\,d\rho \right)\Pi _{n}\sin \omega \,d\omega \,d\phi \right]\\&+\Sigma _{0}^{\infty }\left[r^{n}\int _{'}\int _{'}\left(\int _{r}^{u}{\frac {g{\bar {\omega }}}{\rho ^{n-1}}}\right)\Pi _{n}\sin \omega \,d\omega \,d\phi \right].\end{aligned}}$

The double integral $\int ^{'}\int ^{'}$ is to be extended only to the points in respect to which the radius $u$ from the surface of the molecule is $<r$, and the integral $\int _{'}\int _{'}$ is to be extended to the points in respect to which $u>r$.

By means of a beautiful theorem which M. Poisson has demonstrated in the Memoir already quoted, and in the additions to the Connaissance des Temps for the year 1831, the functions given by the integrals

$\int _{0}^{r}g{\bar {\omega }}\rho ^{n}+2\,d\rho ,\qquad \int _{r}^{u}g{\bar {\omega }}\rho ^{n}+2\,d\rho ,\qquad \int _{r}^{u}{\frac {g{\bar {\omega }}}{\rho ^{n-1}}}\,d\rho$

may be represented by series of integer and rational functions of the spherical co-ordinates. Let $\Sigma H_{n}$, $\Sigma H_{n}^{'}$, $\Sigma H_{n}^{''}$, be these series; if the functions $H_{n}^{'}$, $H_{n}^{''}$ shall be found, so that they may be discontinued, and such that they are reduced to zero, the first for all the values of $u<r$, and the second for the values of $u<r$, we shall be able by means of the known theorems to reduce the expression for $G$ to the form

$G=\Sigma _{0}^{\infty }{\frac {4\pi }{2n+1}}\left({\frac {1}{r^{n+1}}}H_{n}+{\frac {1}{r^{n+1}}}H_{n}^{'}+r^{n}H_{r}^{''}\right).$

Such are the expressions for $F$ and $G$ which should be introduced into the equation (III). We might directly employ those which give the values of $G$, because they are always determinable when the position, figure, and density of the molecules are known; but the same thing cannot be done with the expression for $F$. This integral includes the quantity $q$, which is still unknown; and we should not be able to determine it by the condition that it would render the equation (III) identical without previously performing the integrations, an operation which would require the same function to be known. In order to avoid this difficulty, we are about to employ for the purpose of determining $q$ a differential equation corresponding with that marked (III), but in which the density $q$ is no longer included under the signs of integration.

The sum of these equations (I)', when they are differentiated in reference to $x$, $y$, $z$, respectively, gives

(VI.)

$k\left({\frac {d^{2}q}{dx^{2}}}+{\frac {d^{2}q}{dy^{2}}}+{\frac {d^{2}q}{dz^{2}}}\right)=4\pi fq.$

and it being observed that

${\frac {d^{2}F}{dx^{2}}}+{\frac {d^{2}F}{dy^{2}}}+{\frac {d^{2}F}{dz^{2}}}=-4\pi fq,\qquad \qquad {\frac {d^{2}G}{dx^{2}}}+{\frac {d^{2}G}{dy^{2}}}+{\frac {d^{2}G}{dz^{2}}}=0$

with respect to which see the third volume of the Bulletin de la Société Philomatique, p. 388.

If in this equation we change the differentials taken relatively to the rectangular co-ordinates into differentials taken relatively to the polar co-ordinates, we have

(1)

$k\left\{{\frac {d^{2}rq}{dr^{2}}}+{\frac {1}{r^{2}\sin \theta }}{\frac {d\left(\sin \theta {\frac {drq}{d\theta }}\right)}{d\theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {d^{2}rq}{d\psi ^{2}}}\right\}=4\pi frq.$

Let us suppose that $r\,q$ is developed in a series of integer and rational functions of the spherical co-ordinates, so that we may have

(2)

$rq=Q_{0}+Q_{1}+Q_{2}\ldots \ldots +Q_{i}+\mathrm {etc.} ;$

in which any one of the quantities $Q_{i}$ renders identical the equation

(3)

${\frac {d\left(\sin \theta {\frac {dQ_{i}}{d\theta }}\right)}{\sin \theta \,d\theta }}+{\frac {1}{\sin ^{2}\theta }}{\frac {d^{2}Q_{i}}{d\psi ^{2}}}+i(i+1)Q_{i}=0$

On this supposition the equation (1) will be satisfied by taking in general

(4)

$k\left\{{\frac {d^{2}Q_{i}}{dr^{2}}}-{\frac {i(i+1)}{r^{2}}}Q_{i}\right\}=4\pi fQ_{i}.$

In order to integrate this differential equation of the second order^[1] let us take

$Q_{i}={\frac {Q_{i}^{(1)}}{r}}-{\frac {1}{i}}{\frac {dQ_{i}^{(1)}}{dr}},$

and consequently

${\begin{aligned}{\frac {dQ_{i}}{dr}}&=-{\frac {Q_{i}^{(1)}}{r^{2}}}+{\frac {1}{r}}{\frac {dQ_{i}^{(1)}}{dr}}-{\frac {1}{i}}{\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}\\{\frac {d^{2}Q_{i}}{dr^{2}}}&=2{\frac {Q_{i}^{(1)}}{r^{3}}}-{\frac {2}{r^{2}}}{\frac {dQ_{i}^{(1)}}{dr}}+{\frac {1}{r}}{\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}-{\frac {1}{i}}{\frac {d^{3}Q_{i}^{(1)}}{dr^{3}}}\end{aligned}}$

By making the substitutions in the equation (4) we shall be able to exhibit the result in the following form:

$k\left[{\frac {1}{r}}\left\{{\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}-{\frac {(i-1)i}{r^{2}}}Q_{i}^{(1)}\right\}-{\frac {1}{i}}\left\{{\frac {d^{3}Q_{i}^{(1)}}{dr^{3}}}-{\frac {(i-1)i^{(1)}}{r^{2}}}{\frac {dQ_{i}^{(1)}}{dr}}+{\frac {2i(i-1)}{r^{2}}}Q_{i}^{(1)}\right\}\right]={\frac {4\pi f}{r}}Q_{i}^{(1)}-{\frac {4\pi f}{i}}{\frac {dQ_{i}^{(1)}}{dr}}.$

The foregoing equation is satisfied by taking

$k\left({\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}-{\frac {(i-1)i}{r^{2}}}Q_{i}^{(1)}\right)=4\pi fQ_{i}^{(1)}.$

This equation is of the same form with that proposed (4), except only that $i$ is replaced by $i-1$. If therefore again, in the latter, we put

$Q_{i}^{(1)}={\frac {1}{r}}Q_{i}^{(2)}-{\frac {1}{i-1}}{\frac {dQ_{i}^{(2)}}{dr}}$

we shall deduce from it another in terms of $Q_{i}^{(2)}$, in which $i-1$ will be replaced by $i-2$; and by continuing these substitutions we shall finally obtain the equation

$k{\frac {d^{2}Q_{i}^{(i)}}{dr^{2}}}=4\pi fQ_{i}^{(i)};$

which is integrable by the known methods, and gives

$Q_{i}^{(i)}=T_{i}e^{r{\sqrt {\frac {4\pi f}{k}}}}+V_{i}e^{-r{\sqrt {\frac {4\pi f}{k}}}}$

where $T_{i}$ and $V_{i}$ may be considered as two arbitrary functions of $\theta$ and $\psi$ of the order $i$ which satisfy the equation (3).

By adopting this value $Q_{i}^{(i)}$, and by afterwards taking

$Q_{i}^{(i-1)}={\frac {1}{r}}\quad Q_{i}^{(i)}\quad \;-\;\;\;\;{\frac {dQ_{i}^{(i)}}{dr}}$

$Q_{i}^{(i-2)}={\frac {1}{r}}\quad Q_{i}^{(i-1)}\;-\;{\frac {1}{2}}{\frac {dQ_{i}^{(i-1)}}{dr}}$

$Q_{i}^{(i-3)}={\frac {1}{r}}\quad Q_{i}^{(i-2)}\;-\;{\frac {1}{3}}{\frac {dQ_{i}^{(i-2)}}{dr}}$

$\dots \dots \dots$

$\dots \dots \dots$

$\dots \dots \dots$

$Q_{i}={\frac {1}{r}}Q_{i}^{(1)}-{\frac {1}{i}}{\frac {dQ_{i}^{(1)}}{dr}}$

the last of these quantities will satisfy the equation (4), and will be its complete integral.

If the successive substitutions are performed, and, for brevity's sake, we make

$a_{i}^{(i)}={\frac {(-1)^{i}}{1.2.3..i}};\qquad a_{i}^{(i-\prime )}=\left[-{\frac {1}{i}}\right]^{i}\Sigma {\frac {\left(1+{\frac {\prime -1}{i+1}}\right)a_{(i-1)}^{(i-\prime )}}{\left[-{\frac {1}{i+1}}\right]^{i+1}}}\colon$[2]

which gives, in the particular case of $\prime =i$, $a_{i}^{(0)}=\left[1+{\frac {i-1}{i}}\right]^{i}$, we shall have

$Q_{i}={\frac {a_{i}^{(0)}}{r^{i}}}Q_{i}^{(i)}+{\frac {a_{i}^{(1)}}{r^{i-1}}}{\frac {dQ_{i}^{(i)}}{dr}}+{\frac {a_{i}^{(2)}}{r^{i-2}}}{\frac {d^{2}Q_{i}^{(1)}}{dr^{2}}}\cdots \cdots \cdots +a_{i}^{(i)}{\frac {d^{i}Q_{i}^{(i)}}{dr^{i}}}.$

Now if we make

${\begin{aligned}\Omega _{i}(r')&=\left\{{\frac {a_{i}^{(0)}}{r^{\prime i}}}+{\frac {a_{i}^{(i)}}{r^{\prime i-1}}}\alpha +{\frac {a^{(2)}}{r^{\prime i-1}}}\alpha ^{2}\cdots \cdots +{\frac {a_{i}^{(i)}}{r^{\prime 0}}}\alpha ^{i}\right\}e^{\alpha r'},\\\Omega '_{i}(r')&=\left\{{\frac {a_{i}^{(0)}}{r^{\prime i}}}-{\frac {a_{i}^{(1)}}{r^{\prime i-1}}}\alpha +{\frac {a_{i}^{(2)}}{r^{\prime i-2}}}\alpha ^{2}\cdots \cdots +{\frac {a_{i}^{(i)}}{r^{\prime 0}}}\alpha ^{i}\right\}e^{-\alpha r'},\end{aligned}}$

where $\alpha$ is put instead of ${\sqrt {\frac {4\pi f}{k}}}$ we shall have

$Q_{i}^{'}=T_{i}^{'}\Omega _{i}(r')+V_{i}^{'}\Omega '(r'),$

and the expression for $F$ may take the form

(5)

${\begin{aligned}F&=\Sigma _{0}^{\infty }\iint f\left\{{\frac {1}{r^{\prime n+1}}}\int _{0}^{r}\Sigma _{0}^{\infty }\Omega _{i}(r')T'_{i}r^{\prime n+1}\,dr'\right.\\&+\left.r^{n}\int _{0}^{\infty }\Sigma _{0}^{\infty }{\frac {\Omega _{i}(r')T'_{i}\,dr'}{r^{\prime n}}}\right\}P_{n}\sin \theta '\,d\theta '\,d\psi '\end{aligned}}$

$=\Sigma _{0}^{\infty }\iint f{\bigg \lbrace }{\frac {1}{r^{''+1}}}\int _{0}^{r}\Sigma _{0}^{\infty }\Omega _{i}^{'}(r')V_{i}^{'}r^{'n+1}\,dr'+r^{n}\int _{r}^{\infty }\Sigma _{0}^{\infty }{\frac {\Omega _{i}^{'}(r')V_{i}^{'}\,dr'}{r^{'n}}}{\bigg \rbrace }P_{n}\sin \theta '\,d\theta '\,d\psi '.$

The functions $T'$, $V_{i}^{'}$ of this expression remain arbitrary; and, as the sum of an infinite number of these fund ions may be employed to represent any function whatsoever, they will serve as two arbitrary functions which are to complete the integral of the equation (1).

When in some particular cases the integrations of the preceding formula shall have been performed by substituting its expression in the equation (III), the functions $T_{i}$ and $V_{i}$ will be determined by comparing them with those of the same order introduced by means of the different expressIons for $G$; so that this equation may become identical. All being thus determined, the densty $q$ given by the formula (2) will be known.

We have hitherto left our formula in all their generality, so that one may be the better able to judge of the restrictions to which we shall subject them while making the first applications of them. In the present state of our physical knowledge, the figure of the material molecules is totally unknown. We will therefore begin by considering the most simple case,—that in which their form is spherical, and their density uniform. We will, besides, assign to these molecules a very small volume, and suppose them in their state of equilibrium at a mutual distance, which is very considerable as compared with their dimensions. This manner of considering the constitution of bodies has been adopted by several philosophers as that which is most conformable to truth, and presents at the same time a considerable advantage in an analytical point of view. In adopting it we shall be able, by approximation, to consider the æther as if it were continuously diffused in all directions; and to disregard, in the integration of the formula (5), the small spaces occupied by the material molecules. But as, by proceeding in this manner, we should include in the repulsion of the æther a surplus which is due to the actions answering to the points of space which are really occupied by the molecules, we shall compensate for this surplus by adding to the action of each molecule an action equal and contrary to that of a quantity of æther of the same volume as the molecule, and of the same density as that which answers to the point of space which the molecule occupies. This is done by substituting $g{\bar {\omega }}+f\mathrm {q}$ for $g{\bar {\omega }}$ in the expression for $G$ ($\mathrm {q}$ representing the density which the æther would have at the point occupied by the molecule, and within so small a space we will suppose that density constant), and by extending the integrals of the formula (5) from $\theta '=0$ to $\theta '=\pi$, from $\psi '=0$ to $\psi '=2\pi$, and from $r=0$ to $r=\infty$.

Let us begin with performing the integrations of the formula (V). In consequence of the quantity $g{\bar {\omega }}+fq$ being considered as constant, and as the spherical form of the molecules renders $\rho$ independent of $\omega$ and $\phi$, all the terms of the second and third line of this formula will vanish, and it being observed that we always have

$\int _{0}^{\pi }\int _{0}^{2\pi }\Pi _{n}\sin \omega \,d\omega \,d\phi =0,$

unless in the case of $\Pi ^{0}=1$, which gives

$\int _{0}^{\pi }\int _{0}^{2\pi }\Pi _{0}\sin \omega \,d\omega \,d\phi =4\pi ;$

the expression for $G$ will become $G={\frac {4\pi }{3}}{\frac {(g{\bar {\omega }}+fq)\delta ^{3}}{r}}$, $\delta$ representing the semidiameter of the molecule.

This integral has been obtained under the supposition that the origin of the coordinates is in the centre of the molecule; but the origin may be transferred to any point whatever, by restoring, instead of $r$, its general expression, and writing

(V)'

$G={\frac {4\pi (g{\bar {\omega }}+fq)\delta ^{3}}{3\{(x-\mathrm {x} )^{2}+(y-\mathrm {y} )^{2}+(z-\mathrm {z} )^{2}\}^{\frac {1}{2}}}};$

where $\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$ represent the coordinates of the centre of the molecule.

Before we proceed to the expression for $F$, we had better clearly define the signification of the term $q$ which it contains. We must consider this quantity $(q)$ such as it is given by the equation (III), not as the entire value of the density of the æther, but as the value only of its excess or deficiency above or below the sensibly uniform density which the æther diffused in equilibrium is supposed to have in that part of space. If we represent the latter density by $q_{0}$, the equations (III) and (VI), while we suppress the terms due to the quantities $G$, $G_{1}$, $G_{2}$, &c., must be satisfied by the substitution of $q=q_{0}$: and that, in order that the æther may remain in equilibrium spontaneously, or in consequence of the action of the forces not expressed, whose centres must be supposed to be at a very great distance. If, therefore, we take the difference between the equations resulting from this substitution and the original equations themselves, we shall have

(III)'

$k(q-q_{0})=-F+G+G_{1}+G_{2}\ldots \ldots +G_{\nu }+\mathrm {\&c.}$

${\frac {d^{2}(q-q_{0})}{dx^{2}}}+{\frac {d^{2}(q-q_{0})}{dy^{2}}}+{\frac {d^{2}(q-q_{0})}{dz^{2}}}=4kf(q-q_{0})$

provided that, in $F$, we substitute for $q$ the value of $q-q_{0}$ resulting from this last equation, that is to say, that which is given by the formula

(2)′

$r(q-q_{0})=Q_{0}+Q_{1}+Q_{2}\ldots \ldots +Q_{i}+\mathrm {\&c.}$

This being premised let us return to the formula (5). As the integrations indicated in the second member of this equation may, according to what we have stated at the commencement of this paragraph, be extended from $r'=0$ to $r'=\infty$, $\theta '=0$ to $\theta '=\pi$, and $\psi '=0$ to $\psi '=2\pi$, and as all these limits are independent of each other, observing that we have in general

${\begin{aligned}&\int _{0}^{\pi }\int _{0}^{2\pi }P_{n}T_{i}^{'}\sin \theta '\,d\theta '\,d\psi '=0\\&\int _{0}^{\pi }\int ^{2\pi }P_{n}V_{i}^{'}\sin \theta '\,d\theta '\,d\psi '=0\end{aligned}}$

and in particular when $i=n$;

${\begin{aligned}&\int _{0}^{\pi }\int _{0}^{2\pi }P_{n}T_{n}^{'}\sin \theta '\,d\theta '\,d\psi '={\frac {4\pi }{2n+1}}T_{n};\\&\int _{0}^{\pi }\int _{0}^{2\pi }P_{n}V_{n}^{'}\sin \theta '\,d\theta '\,d\psi '={\frac {4\pi }{2n+1}}V_{n}.\end{aligned}}$

we shall find

${\begin{aligned}F&=\Sigma _{0}^{\infty }{\frac {4\pi }{2n+1}}{\frac {f}{r^{n+1}}}\left\{T_{n}\int _{0}^{r}\Omega _{n}(r^{\prime })r^{\prime n+1}\,dr^{\prime }\right.\\&\left.\qquad +V_{n}\int _{0}^{r}\Omega _{n}^{\prime }(r^{\prime })r^{\prime n+1}\,dr^{\prime }\right\}\\&+\Sigma _{0}^{\infty }{\frac {4\pi }{2n+1}}fr^{n}\left\{T_{n}\int _{0}^{\infty }{\frac {\Omega _{n}(r^{\prime })}{r^{\prime n}}}dr^{\prime }\right.\\&\qquad +\left.V_{n}\int _{r}^{\infty }{\frac {\Omega _{n}^{\prime }(r^{\prime })}{r^{\prime n}}}dr^{\prime }\right\}\end{aligned}}$

Without actually making the substitutions of the expressions previously given for $G$, and latterly for $F$, in the equation (III)' for the purpose of comparing the functions of the spherical coordinates of the same degree which are to render it identical, we see that, as $G$, $G_{1}$, $G_{2}$, &c., contain none of these functions, all the $T_{n}$ and $V_{n}$ must be null, with the exception of $T_{0}$ and $V_{0}$, which answer to the value $n=0$, and represent two arbitrary constants.

The expression for $F$ will then be reduced to

(5)′

${\begin{aligned}F&=4\pi f{\frac {1}{r}}\int _{0}^{r}(T_{0}e^{\alpha r^{\prime }}+V_{0}e^{-\alpha r^{\prime }})r'\,dr'\\&{}+4\pi f\int _{r}^{\infty }(T_{0}e^{\alpha r^{\prime }}+V_{0}e^{-\alpha r^{\prime }})dr'\end{aligned}}$

All the quantities $T_{n}$ and $V_{n}$ being null, except $T_{0}$ and $V_{0}$, the values of $Q_{n}$ will also be null, except that of $Q_{0}$: the formula (2)′ will then give $q-q_{0}={\frac {T_{0}e^{\alpha r}+V_{0}e^{-\alpha r}}{r}}$

When $r=\infty$ we must have $q=q_{0}$; we must then also have $T_{0}=0$, and there will remain only $q=q_{0}+{\frac {V_{0}}{r}}e^{-\alpha r}$.

By performing the integrations of the formula (5)′ within the limits indicated, and observing that $T_{0}=0$, we shall obtain

$F=-k{\frac {V_{0}}{r}}\left(e^{-\alpha r}-1\right);$

As, in the differential expression for $F$, we may change $x'$ into $x'-\mathrm {x}$, and x into $x-\mathrm {x}$, without any change taking place in its value, and as a similar change may be made in respect to the other coordinates, it follows that, by taking the point $\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$, as the origin of the coordinates, we shall be able, in the two preceding formulas, to put

$r={\sqrt {(x-\mathrm {x} ^{2}+(y-\mathrm {y} ^{2}+(z-\mathrm {z} ^{2}}}$

or, generally,

$r_{\nu }={\sqrt {(x-\mathrm {x} _{\nu })^{2}+(y-\mathrm {y} _{\nu })^{2}+(z-\mathrm {z} _{\nu })^{2}.}}$

Now if, by placing the origin of the coordinates in the centre of each molecule respectively, we substitute these expressions of $F$ and $q$, and that previously found for $G$ in the equation (III)′, and take successively for $V_{0}$ as many constants as there are molecules, we shall find that the equation

$\Sigma kV_{0}^{(\nu )}{\frac {e^{-\alpha r_{\nu }}}{r_{\nu }}}=\Sigma kV_{0}^{(\nu )}{\frac {e^{-\alpha r_{\nu }}-1}{r_{\nu }}}+\Sigma {\frac {4\pi (g{\bar {\omega }}_{\nu }+f\mathrm {q} _{\nu })\delta _{\nu }^{3}}{3r_{\nu }}}$

will be satisfied by taking for each molecule

$V_{0}^{(\nu )}={\frac {4\pi }{3k}}(g{\bar {\omega }}_{\nu }+f\mathrm {q} _{\nu })\delta _{\nu }^{3}.$

By substituting for $V_{0}^{(\nu )}$ the value just found, we shall finally have

(IV)′

$F=-{\frac {4\pi }{3}}\Sigma (g{\bar {\omega }}_{\nu }+f\mathrm {q} _{\nu })\delta _{\nu }^{3}{\frac {e^{-\alpha r_{\nu }}-1}{r_{\nu }}}$

(III)″

$q=q_{0}+{\frac {4\pi }{3k}}\Sigma (g{\bar {\omega _{\nu }}}+fq_{\nu })\delta _{\nu }^{3}{\frac {e^{-\alpha r_{\nu }}}{r_{\nu }}}.$

where the sums $\Sigma$ are to be extended to all the molecules, including the first.

This last equation determines what the density of the æther must be at each point $x\,y\,z$, in order that it may be in equilibrium when it is submitted to the action of the spherical molecules of matter. The value of this density consists of different terms, each of which is due to a particular molecule, and represents its proper atmosphere. As the quantity of æther diffused through the immensity of space may be considered as infinite, the atmosphere formed by each molecule for itself is always the same, and its density is only superadded to that which the æther in the same places owes to other causes. According to the nature of the molecular actions, the value of the coefficient $\alpha ={\sqrt {\frac {4\pi f}{k}}}$ should be considered as very great: hence it follows that the density of each atmosphere will be incomparably greater when quite near or in contact with the molecule, and will decrease very rapidly as its distance from the molecule increases. This circumstance enables us to determine with ease, by approximation, the value of $q_{\nu }$, or the density of the æther at the surface of any molecule whatsoever, on the supposition that the molecules are not too near each other. If, for instance, we make $r=\delta$ in the term answering to the first molecule, and $r_{1}=\mathrm {r_{1}}$, $r_{2}=\mathrm {r_{2}}$ … $r_{\nu }=\mathrm {r_{\nu }}$ in the other terms, all these will be very small in comparison with the first, and by neglecting them we shall have very nearly

$\mathrm {q} =q_{0}+{\frac {4\pi }{3k}}(g{\bar {\omega }}+f\mathrm {q} )\delta ^{2}$

whence we derive

(6)

$\mathrm {q} ={\frac {q_{0}+{\frac {4\pi }{3k}}g{\bar {\omega }}\delta ^{2}}{1+{\frac {4\pi }{3k}}f\delta ^{2}}}.$

6. We are now in a condition to consider the equilibrium of any molecule whatever, such as it is given by the equations (II).

The quantity $\epsilon$ under the double integral in these equations must be replaced by ${\frac {1}{2}}kq^{2}$. Let us represent the coordinates $x$, $y$, $z$, so far as they belong to the points in contact with the surface of the molecule, by $\mathrm {x} +\xi$, $\mathrm {y} +\eta$, $\mathrm {z} +\zeta$; $\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$ being the coordinates of its centre: by developing the expression for $q$, and stopping, because of the smallness of the molecule, at the first terms, we shall be able to take

$q^{2}=\mathrm {q} ^{2}+2\mathrm {q} {\frac {d\mathrm {q} }{d\mathrm {x} }}\xi +2\mathrm {q} {\frac {d\mathrm {q} }{d\mathrm {y} }}\eta +2\mathrm {q} {\frac {d\mathrm {q} }{d\mathrm {z} }}\zeta .$

If this expression for $q^{2}$ be put in the integral ${\frac {1}{2}}k\int q^{2}\,d\eta \,d\zeta$, and the limits extended to the whole surface of the molecule, it is easy to see that it is reduced to $k\mathrm {q} {\frac {d\mathrm {q} }{d\mathrm {x} }}\iint \xi \,d\eta \,d\zeta$. But $\iint \xi \,d\eta \,d\zeta$, expresses the volume $v$ of the molecule, which is equal to ${\frac {4\pi }{3}}\delta ^{3}$; the term on the right in the first of the equations (II) will therefore be simply represented by $kv\mathrm {q} {\frac {d\mathrm {q} }{d\mathrm {x} }}$. It is proper to remark, that in the value of $q{\frac {d\mathrm {q} }{d\mathrm {x} }}$, we are not to include the term which, in the expression for $q$ marked (III)″ is due to the molecule whose equilibrium we are considering, because this term undergoes a change of sign at the two opposite sides of the surface of the molecule, and vanishes within the limits between which the integral is extended.

The inspection of the triple integral which gives the value $\Phi$ is sufficient to show that this integral must be given by the same function that represents $F$, in which $f$, $x$, $y$, $z$ may be replaced by $g$, $\xi$, $\eta$, $\xi$. If, because of the smallness of the dimensions of the molecule, we consider in the differential ${\frac {d\Phi }{d\xi }}$, the coordinates $\xi$, $\eta$, $\zeta$, which answer to any point of the surface as being constant, and substitute for them $\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$ which answer to the centre, then, it being observed that $\iiint d\xi \,d\eta \,d\zeta$, represents the volume $v$ of the molecule, the first integral of the second member of the first of the equations (II) may be represented ${\bar {\omega }}{\frac {d\Phi }{d\mathrm {x} }}$.

The value of $\Phi$ being deduced from the expression for $F$, such as it is given by the equation (IV)′, will contain, as we have already observed, a surplus of action, due to the æther which is supposed to occupy the place of the molecules also. It will therefore be necessary to make a compensation here also, by adding to the contrary action of the molecules an equal quantity; that is to say, by changing in the triple integral represented by $\Gamma _{\nu }$ the mass $\gamma {\bar {\omega }}$ into the mass $\gamma {\bar {\omega }}_{\nu }+g\mathrm {q} _{\nu }$. If we conceive this change made, the expression for $\Gamma _{\nu }$ will be of the same form as that for $G$ marked (V)′, except that $x$, $y$, $z$ and $g{\bar {\omega }}+f\mathrm {q}$ will be replaced by $\xi$, $\eta$, $\zeta$, and $\gamma {\bar {\omega }}_{\nu }+g\mathrm {q} _{\nu }$, and $\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$ by $\mathrm {x} _{\nu }$, $\mathrm {y} _{\nu }$, $\mathrm {z} _{\nu }$. Let us then, by approximation, introduce into the differential ${\frac {d\Gamma }{d\xi }}$ instead of the coordinates ($\xi$, $\eta$, $\zeta$) of the surface, the coordinates ($\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$) of the centre considered as constant; if we perform the integration, which is done by substituting the volume $v$ for $\iiint d\,\xi \,d\eta \,d\zeta$, the term which stands under the sign $\Sigma$ in the first of the equations (II) will be represented by ${\bar {\omega }}{\frac {d\,\Gamma _{\nu }}{d\mathrm {x} }}$.

If we now write in their places all the expressions just found for the integrals which constitute the first of the equations (II), we shall have

$kqv{\frac {d\,\mathrm {q} }{d\,\mathrm {x} }}={\bar {\omega }}v{\frac {d\Phi }{d\mathrm {x} }}-{\bar {\omega }}v\Sigma {\frac {d\Gamma _{\nu }}{d\mathrm {x} }}.$

By similar substitutions the second and the third equation will give respectively

${\begin{aligned}k\mathrm {q} v{\frac {d\mathrm {q} }{d\mathrm {y} }}&={\bar {\omega }}v{\frac {d\Phi }{d\mathrm {y} }}-{\bar {\omega }}v\Sigma {\frac {d\Gamma _{\nu }}{d\mathrm {y} }}\\k\mathrm {q} v{\frac {d\mathrm {q} }{d\mathrm {z} }}&={\bar {\omega }}v{\frac {d\Phi }{d\mathrm {z} }}-{\bar {\omega }}v\Sigma {\frac {d\Gamma _{\nu }}{d\mathrm {z} }}.\end{aligned}}$

These three equations must hold good for the particular values $\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$; $\mathrm {x} _{1}$, $\mathrm {y} _{1}$, $\mathrm {z} _{1}$;… … …$\mathrm {x} _{\nu }$, $\mathrm {y} _{\nu }$, $\mathrm {z} _{\nu }$, &c., which answer to the centre of the molecules in their state of equilibrium; and as each molecule furnishes three similar equations, the whole collectively will be sufficient to enable us to determine the unknown quantities.

If from the formulæ marked (III)″, (IV)′, (V)′ we derive, by means of the changes already indicated, the expressions for ${\frac {d\mathrm {q} }{d\mathrm {x} }}$, ${\frac {d\Phi }{d\mathrm {x} }}$, ${\frac {d\Gamma _{\nu }}{d\mathrm {x} }}$, we find

${\begin{aligned}{\frac {d\mathrm {q} }{d\mathrm {x} }}&=-{\frac {1}{k}}\Sigma v_{\nu }(g{\bar {\omega }}_{\nu }+f\mathrm {q} _{\nu }){\frac {(1+\alpha r_{\nu })e^{-\alpha r_{\nu }}}{r_{\nu }^{2}}}\centerdot {\frac {\mathrm {x} _{\nu }-\mathrm {x} }{r_{\nu }}}\\{\frac {d\Phi }{d\mathrm {x} }}&={\frac {g}{f}}\Sigma v_{\nu }(g{\bar {\omega }}_{\nu }+f\mathrm {q} _{\nu })\left\{{\frac {(1+\alpha r_{\nu }e^{-\alpha r_{\nu }}}{r_{\nu }^{2}}}-{\frac {1}{r_{\nu }^{2}}}\right\}{\frac {\mathrm {x} _{\nu }-\mathrm {x} }{r_{\nu }}}\\{\frac {d\Gamma _{\nu }}{d\mathrm {x} }}&=-v^{\nu }(\gamma {\bar {\omega }}_{\nu }+g\mathrm {q} _{\nu }){\frac {1}{r_{\nu }^{2}}}\centerdot {\frac {\mathrm {x} _{\nu }-\mathrm {x} }{r_{\nu }}}\end{aligned}}$

and shall obtain ${\frac {d\mathrm {q} }{d\mathrm {y} }}$, ${\frac {d\Phi }{d\mathrm {y} }}$, ${\frac {d\Gamma _{\nu }}{d\mathrm {y} }}$; ${\frac {d\mathrm {q} }{d\mathrm {z} }}$, changing in these formulæ $\mathrm {x}$ into $\mathrm {y}$ and into $\mathrm {z}$.

If we introduce these expressions into the foregoing equations, recollecting that, according to the hypothesis of Franklin and Æpinus, we must make $f=g$, and take $\gamma$ a little less than $g$, the result will be

(A)

${\begin{aligned}gv(\mathrm {q} &+{\bar {\omega }})\Sigma v_{\nu }({\bar {\omega }}+\mathrm {q} _{\nu })(1+\alpha r_{\nu })e^{-\alpha r_{\nu }}{\frac {\mathrm {x} _{\nu }-\mathrm {x} }{r_{\nu }^{3}}}\\&-(g-\gamma ){\bar {\omega }}v\Sigma {\bar {\omega }}_{\nu }v_{\nu }{\frac {\mathrm {x} _{\nu }-\mathrm {x} }{r_{\nu }^{3}}}=0\\gv(\mathrm {q} &+{\bar {\omega }})\Sigma v_{\nu }({\bar {\omega }}_{\nu }+\mathrm {q} _{\nu })(1+\alpha r_{\nu })e^{-\alpha r_{\nu }}{\frac {\mathrm {y} _{\nu }-\mathrm {y} }{r_{\nu }^{3}}}\\&-(g-\gamma ){\bar {\omega }}v\Sigma {\bar {\omega }}_{\nu }v_{\nu }{\frac {\mathrm {y} _{\nu }-\mathrm {y} }{r^{3}}}=0\\gv(\mathrm {q} &+{\bar {\omega }})\Sigma v_{\nu }({\bar {\omega }}_{\nu }+\mathrm {q} _{\nu })(1+\alpha r_{\nu })e^{-\alpha r_{\nu }}{\frac {\mathrm {z} _{\nu }-\mathrm {z} }{r_{\nu }^{3}}}\\&-(g-\gamma ){\bar {\omega }}v\Sigma {\bar {\omega }}_{\nu }v_{\nu }{\frac {\mathrm {z} _{\nu }-\mathrm {z} }{r_{\nu }^{3}}}=0,\end{aligned}}$

where the sums $\Sigma$ are to be extended to all the members $\nu$, that is to say, to all the molecules except that whose equilibrium we are considering.

7. The equations which we have just found are those which must take place in case of equilibrium, or in the natural state of a body composed of spherical molecules, if Franklin's hypothesis respecting statical electricity may be applied to the constitution of bodies also. The form in which the equations present themselves shows that this equilibrium takes place exactly as if there existed between each pair of molecules a reciprocal action, in the direction of the straight line which would join their centres of gravity, and would be represented by

(a)

$gv({\bar {\omega }}+\mathrm {q} )v_{1}({\bar {\omega }}_{1}+\mathrm {q} _{1}){\frac {(1+\alpha r_{1})e^{-\alpha r^{2}}}{r_{1}^{2}}}-(g-\gamma )v{\bar {\omega }}.v_{1}{\bar {\omega }}_{1}{\frac {1}{r_{1}^{2}}}.$

Let us examine the nature of this action. We are able to distinguish in its expression the products $gv({\bar {\omega }}+\mathrm {q} ).v_{1}({\bar {\omega }}_{1}+\mathrm {q} _{1})$, $(g-\gamma ){\bar {\omega }}v.{\bar {\omega }}_{1}v_{1}$, the constant $\alpha$ and the variable $r_{1}$.

As the difference $(g-\gamma )$ between these two accelerative forces is to be supposed very small relatively to $g$, the product of this force by the masses $v({\bar {\omega }}+q)v_{1}({\bar {\omega }}_{1}+\mathrm {q} _{1})$ will, for a twofold reason, be greater than the product of the difference $g-\gamma$ by the masses ${\bar {\omega }}v.{\bar {\omega }}v_{1}$.

The value $\alpha$ depends on that of $f$ and $k$, that is to say on the repulsive force of the atoms of the æther, their mutual distances, their masses, and their volumes, which are all unknown to us. The agreement of the results of calculation with those of experiment requires that $\alpha$ should be a very high number.

On the condition that $a$ is very great, the first term of the expression (a) will decrease rapidly with $r_{1}$, because of the multiplier $e^{-\alpha r_{1}}$; if then $r_{1}$ has a greater value than that which renders this expression null, the force represented by the last term will preponderate over that represented by the first; and if $r_{1}$ be so great that this term may be neglected as of no value, then the only remaining force will be that given by the last term. This term being negative, the force which corresponds with it tends to bring the molecules nearer to each other; and as it is in the direct ratio of the product of the masses, and the inverse ratio of the square of the distance, it will exactly represent the universal gravitation which takes place at finite distances.

By diminishing $r_{1}$ we shall obtain a value that will satisfy the equation

(b)

$gv({\bar {\omega }}+\mathrm {q} )v_{1}({\bar {\omega }}_{1}+\mathrm {q_{1}} ){\frac {(1+\alpha r_{1})e^{-\alpha r_{1}}}{r_{1}^{2}}}-(g-\gamma )v{\bar {\omega }}.v_{1}{\bar {\omega }}_{1}{\frac {1}{r_{1}^{2}}}=0.$

At this distance two molecules would remain in equilibrium, and as the differentiation of this equation gives the result

$-gv({\bar {\omega }}+\mathrm {q} )v_{1}({\bar {\omega }}+\mathrm {q} _{1}){\frac {\alpha ^{2}e^{-\alpha r_{1}}}{r_{1}}}$

which is always negative, the equilibrium will be permanently fixed. Should it be attempted, by the application of an external force, to bring the molecules nearer to each other, the repulsive force represented by the first term of the expression (a), which would now increase in a greater ratio than the attractive force represented by the last term, would produce a resistance to such an approximation: on the other hand, if it should be sought to remove the molecules to a greater distance from each other, the repulsive force would decrease in a greater ratio, and the attractive would preponderate and prevent the separation. These two molecules would therefore be so placed relatively to each other as by mutual adhesion to form a whole, and we should not be able to remove the one without at the same time removing the other. Thus these molecules present a picture in which the hooked atoms of Epicurus are as it were generated by the love and hatred of the two different matters of Empedocles.

As the attractive force is null at the distance which we have been just now considering, and at a greater distance decreases as the square of the distance of the molecules, there must be an intermediate point at which it reaches its maximum. By the ordinary rules of the differential calculus we find that the function (a) is a maximum when

(c)

${\begin{aligned}-&gv({\bar {\omega }}+\mathrm {q} )v_{1}({\bar {\omega }}_{1}+\mathrm {q} _{1})(1+\alpha r_{1}+{\frac {1}{2}}\alpha ^{2}r_{1}^{2})e^{-\alpha r_{1}}+(g-\gamma )\\&v{\bar {\omega }}.v_{1}{\bar {\omega }}=0;\end{aligned}}$

that is to say, that it is at the distance $r_{1}$ we should find, by the resolution of this equation, that the molecules attract each other most forcibly.

Recapitulating these results, we shall say then, that the action of two spherical molecules on each other is repulsive, from their point of contact to a distance given by the equation (b). At this distance the two molecules are in a state of fixed equilibrium, and as it were linked together; at a greater distance their action is attractive, and the attraction continues to increase until they are at the distance $r_{1}$ furnished by the equation (c), which distance is still very inconsiderable because of the magnitude of $\alpha$ in the exponential term $e^{-\alpha r_{1}}$. From this point the force remains always attractive, and, when the distance has acquired a sensible value, follows the inverse ratio of the square of the distance. All these properties of molecular action flow as necessary consequences from Franklin's hypothesis respecting statical electricity, and appear perfectly conformable to those indicated by the phænomena.

Let us suppose four homogeneous and equal molecules placed at the points of a regular tetrahedron. If we assume as the origin of the coordinates the place occupied by the molecule whose equilibrium it is proposed to consider, and as the plane of the $x\,y$, a plane parallel to that in which the three others are found, the coordinates of these molecules will be given by the formulæ

${\begin{aligned}\mathrm {x} &=0&\mathrm {y} &=0&\mathrm {z} &=0\\\mathrm {x} _{1}&={\frac {r}{\sqrt {3}}}\cos \beta &\mathrm {y} _{1}&={\frac {r}{\sqrt {3}}}\sin \beta &\mathrm {z} _{1}&=r{\sqrt {\frac {2}{3}}}\\\mathrm {x} _{2}&={\frac {r}{\sqrt {3}}}\cos \left(\beta +{\frac {2\pi }{3}}\right)&\mathrm {y} _{2}&={\frac {r}{\sqrt {3}}}\sin \left(\beta +{\frac {2\pi }{3}}\right)&\mathrm {z} _{2}&=r{\sqrt {\frac {2}{3}}}\\\mathrm {x} _{3}&={\frac {r}{\sqrt {3}}}\cos \left(\beta +{\frac {4\pi }{3}}\right)&\mathrm {y} _{3}&={\frac {r}{\sqrt {3}}}\sin \left(\beta +{\frac {4\pi }{3}}\right)&\mathrm {z} _{3}&=r{\sqrt {\frac {2}{3}}}\end{aligned}}$

where $r$ denotes the mutual distance of the molecules, which is the same for all; $\beta$ the angle which is formed in the plane of $x\,y$ with the axis of the $x$, by the projection of the straight line drawn from the molecule placed at the origin of the coordinates to the first of the three others; and $\pi$ the semicircumference.

If these values be substituted in the three equations (A), and it is observed that we always have, whatever may be the value of $\beta$,

${\begin{aligned}\cos \beta &+\cos \left(\beta +{\frac {2\pi }{3}}\right)+\cos \left(\beta +{\frac {4\pi }{3}}\right)=0,&\sin \beta &+\sin \left(\beta +{\frac {2\pi }{3}}\right)\\&+\sin \left(\beta +{\frac {4\pi }{3}}\right)=0,\end{aligned}}$

it will be seen that the two first are verified by themselves, and that the third gives, for the determination of $r$,

$gv^{2}({\bar {\omega }}+\mathrm {q} )^{2}{\frac {(1+\alpha r)e^{-\alpha r}}{r^{2}}}-(g-\gamma )v^{2}{\bar {\omega }}^{2}{\frac {1}{r^{2}}}=0.$

If the density of the æther into which the molecules are plunged, or the quantity $q_{o}$, becomes greater, the density q given by the equation (6) will increase also; that value of $r$ which will satisfy the foregoing equation will consequently become greater, and the molecules will fix themselves in equilibrium at a greater distance. We see in this result that the æther performs the functions of caloric, and that it is to its greater or less density we are to ascribe the temperature and volume of the body. For what else, in fact, is an increase or diminution of temperature in respect to a body, than a new state in which its molecules, placed in equilibrium, form, in consequence of their being more or less widely separated, a greater or less volume. It has been known to philosophers since the time of Galileo, who was the first that clearly pointed out this difference, that we are not to confound the sensation which we experience while this new arrangement of the molecules of our body is taking place, with the motion by which it is produced.

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NOTE.

[The readers of this Memoir will doubtless be interested in referring to Dr. Roget's "Treatise on Electricity" in the Library of Useful Knowledge, published March 15th, 1828; the following passage from which was noticed with reference to M. Mossotti's views, by Prof. Faraday in his lecture at the Royal Institution, Jan. 20th of the present year.—Edit.]

"(239.) It is a great though a common error to imagine, that the condition assumed by Æpinus, namely that the particles of matter when devoid of electricity repel one another, is in opposition to the law of universal gravitation established by the researches of Newton; for this law applies, in every instance to which inquiry has extended, to matter in its ordinary state; that is, combined with a certain proportion of electric fluid. By supposing, indeed, that the mutual repulsive action between the particles of matter is, by a very small quantity, less than that between the particles of the electric fluid, a small balance would be left in favour of the attraction of neutral bodies for one another, which might constitute the very force which operates under the name of gravitation; and thus both classes of phænomena may be included in the same law."

1. The integration of this equation with the second member negative has also exercised the ingenuity of the two illustrious geometers Plana and Paoli. See the Memoirs of the Academy of Turin, vol. xxvi., and those of the Italian Society, vol. xx.
2. The brackets are here employed in the same way as in Vandermonde's notation.