The word Attraction is used to denote what we observe when one body approaches another, or tends to approach it, without any apparent impulse or other cause to which the motion can be ascribed.

We have instances of attraction when iron approaches the magnet, when certain bodies are placed near an excited electric, and when a stone falls to the earth. We say likewise that the earth attracts the moon; by this mode of expression meaning no more than that the moon is continually deflected towards the earth, from the rectilinear course which it would otherwise pursue. It is likewise in this sense that we must be understood when we say that the sun attracts all the planets.

In the instances already mentioned, attraction extends to a distance. In other cases it is confined within limits so extremely narrow as to become imperceptible at an interval which cannot be appreciated by the senses. Of this kind is the attraction which takes place between the particles of the same fluid, as is apparent from the round figure of small drops. To this class likewise belongs the attraction between fluid and solid bodies; whence originate the very interesting appearances observed in capillary tubes, and other kindred phenomena. An attraction between the small elementary particles of all solid bodies is manifest from the force with which they cohere or resist an endeavour to separate them. In many cases the intensity of this force is prodigiously great in contact or at the nearest distances; while it ceases to act upon making the smallest separation between the parts. Lastly, chemistry develops innumerable instances of attraction between the molecules of the bodies about which it is conversant; insomuch that it is to this principle, under the name of affinity, that we must ultimately ascribe the various decompositions and new combinations which occur in that science.

All these phenomena, although very different from one another in other respects, have yet this in common, that we observe in certain bodies a tendency to approach one another, and to resist a separation, with some degree of force. The facts are certain, and are attended with no ambiguity; and it is to express these facts that the term attraction is used in physics.

We likewise observe, in some bodies, a tendency to fly off from one another when they are brought near. This is called repulsion.

The word force has, in general, some degree of obscurity. It is used to denote the cause of motion; but we have no direct knowledge of it, and we judge of its intensity by the effect which we suppose it to produce. In all our reasoning concerning forces, it is the changes of motion which we measure and compare together, and which are really the subjects of our thoughts. Attraction and repulsion are forces or principles of motion, known to us only by the phenomena we observe; but the circumstance of their implying action at a distance is an additional source of obscurity in which other kinds of force do not participate.

It certainly is inconceivable that motion should be produced at a distance, when no connection can be traced between the body moved and that which is supposed to produce the motion. We are strongly impressed with the prejudice, that a body cannot act but where it is; and we find difficulty in admitting that the mere presence of two bodies, without the intervention of any mechanical means, can be a satisfactory cause of motion. On this account attraction has been classed by some with the occult qualities of the schools; and the favourers of this doctrine have been reproached with reviving exploded notions in philosophy. Impulse is a principle of motion more familiar to us, and to which we are not disposed to make equal objection. Whenever the communication of motion can be traced to this source, we are satisfied that the effect is justly explained. Hence many philosophers have been of opinion, that impulse is the only cause of motion that can Attraction be admitted in physical science; and many attempts have accordingly been made to reduce to this principle all cases in which distant bodies act on one another. With regard to these attempts, it will be sufficient to remark here, that they are all built upon hypothesis. No evidence is adduced to prove that such things exist as the elastic ether or gravific matter which they set out with supposing. And, as far as such systems have no other object than to obviate the difficulty of action at a distance, this argument alone is sufficient to confute them, without adverting to the difficulties that attend each of them separately, their inconsistency with the received laws of motion, and the innumerable contradictions and improbabilities to which they are liable on every side.

A little reflection is sufficient to show that, in reality, we have no clearer notion of impulse as the cause of motion than we have of attraction. We can as little give a satisfactory reason why motion should pass out of one body into another on their contact, as we can, why one body should begin to move, or have its motion increased, when it is placed near another body. It is equally impossible in both cases to prove that there is a necessary connection between the related facts; and in this respect both the phenomena are alike inexplicable.

When motion is produced by impulse, it is probably the circumstance of contact apparently taking place which leads us to think that the effect is so clearly explained. It is in this manner only, or by actual contact, that we ourselves can move external objects. We have no power of producing motion in distant bodies, except by the intervention of other bodies on which we act immediately. Impulse is, therefore, a cause of motion familiar to us, and strikes us as the plainest and most satisfactory ultimate principle at which we can arrive. On the other hand, when one body attracts another at a distance, there is nothing familiar to us with which we can compare it; our curiosity is excited, and we are led to seek out some hidden connection between them.

But it may be doubted whether there is actual contact in any case of the communication of motion. When a body is impelled by the air, it will hardly be affirmed that the particles of that elastic fluid are in contact with one another, since there is no space, however small, within which a given bulk of it may not be compressed, by applying an external force sufficiently great. The particles of air, therefore, act on one another at a distance; and the same thing must be true of all other elastic fluids. And, by the way, what is here said is sufficient to prove, that no scheme, founded on the hypothesis of an elastic ether, will enable us to account for attraction; because such a contrivance can do nothing more than substitute one species of action at a distance instead of another. There is good reason to think that absolute contact never takes place in the component parts of the hardest and most compact solid bodies. This seems to be an unavoidable consequence of the fact, well established by experience, that all bodies contract in their bulk by cold, and expand by heat. It is therefore not only not impossible, but it is even in some degree probable, that the communication of motion may, in every instance, be a case of action at a distance.

If, then, we are apt to think that impulse is a clearer physical principle than attraction, there is in reality no good ground for the distinction; it has its origin in prejudice, and in our mistaking the proper object of natural philosophy. All our researches in nature are confined to the phenomena we observe, and to the laws by which they are regulated. A physical cause is no other than a general fact discovered by a careful observation and an attentive comparison of many particular and subordinate facts. We have no evidence, independent of experience, that any consequence, deduced from a physical cause, will actually take place. There is in this case no necessary connection from which we can, with absolute certainty, infer the expected event. If, then, we regard impulse and attraction as principles founded in fact, and regulated by laws confirmed by observation and experiment, they are both equally entitled to be classed as physical causes, and they ought both to be admitted as of equal authority in explaining the phenomena of the universe.

If we turn our attention to the different kinds of attraction enumerated above, and inquire what progress has been made in the investigation of their laws of action, we shall find that, generally speaking, this branch of physics has been little advanced. We are very imperfectly acquainted with magnetical and electrical attraction. We know still less of those attractive powers which take place at small distances, and which are confined within such narrow limits that their mode of action escapes the observation of our senses. Attraction is, indeed, much used by philosophers to account for many important natural phenomena; but their explanations are often vague, and destitute of that precision which ought always to be aimed at in physical science. There is only one class of phenomena in which the laws of attraction have been fully developed. We allude to gravitation, that principle which occasions the fall of heavy bodies at the surface of the earth, and which retains the planets and comets in their orbits. Referring the other species of attraction, which are little susceptible of general discussion, to their several heads, we shall now confine our attention to gravitation.

Traces of the principle of gravitation are to be found discovery in writers of great antiquity; but their speculations on this subject do not go beyond a vague notion of a tendency which the planets have to one another, or to a common centre. It would contribute little either to entertainment or instruction to collect all the passages of ancient authors that speak of this principle. The revival of the true system of the world by Copernicus introduced the most admirable simplicity in the explanation of the planetary motions, and likewise led to more just conjectures concerning the laws by which they are upheld. Copernicus himself attributed the round figure of the planets to a tendency which their parts possess of uniting with one another, thus extending to all the planets that attraction which we observe at the surface of the earth. He stopt short indeed at this point, conceiving attraction to be confined to the matter of each planet, without making it extend from one planet to another, so as to actuate all the bodies of the system. This step was made by the bold and systematic genius of Kepler. Adopting the opinion of Dr Gilbert of Colchester, that the earth is a great magnet, Kepler formed to himself a notion of attraction, in some respects remarkably just. He says that the earth and moon attract one another, and, were it not for some powers which retain them in their orbits, they would move towards one another, and would meet in their common centre of gravity. He attributes the tide to the moon's attraction (virtus tractoria quae in luna est), which heaps up the waters of the ocean immediately under her. But in many respects his notions of attraction were fanciful and extravagant; a more perfect knowledge of the laws of motion than had been attained to in his time, and a new geometry, were both wanting in order to guide him in this research without danger of wandering. Yet he was able to penetrate so far into the causes of the planetary motions, as to foresee that they would not long continue latent; and he tells us, he was persuaded that "the full discovery of those mysteries was reserved for the next age, when God..." Attraction would reveal them." So full an exposition of a physical system of the world as is contained in the writings of Kepler could not fail to draw the attention of succeeding philosophers. Many remarks concerning the principle of gravitation are to be found in the writings of Fermat, Roberval, Borelli, and other authors; but no one before Newton entertained so clear and systematic a view of the doctrine of universal gravitation as Dr Robert Hooke. In his work on the motion of the earth, published in 1674, twelve years before the appearance of Newton's Principia, he lays down these three positions as the foundations of his system, viz.:

1st. That all the heavenly bodies have not only a gravitation of their parts to their own proper centre, but likewise that they mutually attract each other within their spheres of action.

2ndly. That all bodies having a simple motion will continue to move in a straight line, unless continually deflected from it by some extraneous force, causing them to describe a circle, or an ellipse, or some other curve.

3rdly. This attraction is so much the greater as the bodies are nearer."

The principle of universal gravitation is here very precisely enunciated. Dr Hooke seems to have clearly perceived that the planetary motions are the result of an attraction towards the sun, and of a rectilinieal motion produced by a projectile force. Not having discovered the law according to which the force diminishes as the distance from the sun increases, he contrived experiments to elucidate his theory. Having suspended a ball by means of a long thread, he placed another ball upon a table immediately under the point of suspension, and he caused the suspended ball to revolve round the stationary one. When the movable ball was pushed laterally with a force properly adjusted to its deviation from the perpendicular, it described an exact circle round the ball on the table; in other cases it described an ellipse, or an oval resembling an ellipse, having the other ball in the centre. Dr Hooke observed, that although this experiment in some measure illustrated the planetary motions, yet it did not represent them accurately; because the ellipses which the planets describe have the sun placed in one focus, and not in the centre. Thus, at the appearance of Newton, many things were known, or rather surmised, that prepared the way for the discovery of the principle which regulates the celestial motions. This does not detract in any degree from the glory of Newton, who, discarding the conjectures of his predecessors, proposed to himself to investigate, with mathematical strictness, the law of the attractive force, and to ascertain with precision its sufficiency to retain the planets in their orbits. He invented a new kind of geometry, which was necessary to enable him to accomplish his purpose. With this help, and by admitting nothing without the sanction of the established principles of Dynamics, he deduced from the motions of the celestial bodies the law of universal gravitation, the most important and the most general truth hitherto discovered by the industry and the sagacity of man, viz. "that all the particles of matter attract one another, directly as their masses, and inversely as the squares of their distances."

Having arrived at a principle which belongs to every part of matter, another inquiry comes into view. Setting out from this principle, it is now necessary to proceed in an inverted order, and deduce from it, by synthetical reasoning, the phenomena which we observe in the universe. The first step in this process is to find out the attractive force of the planets, which arises from the united attractions of their component parts. Two things only are involved in this investigation, viz. the known law of attraction between the particles of matter, and the figure of the attracting bodies. This is a subject of great Attraction importance, and it is connected with some principal points of the system of the world, with the theory of the figure of the planets, that of the tides, and many other phenomena. It is but imperfectly discussed in Newton's immortal work; and there is no part of his philosophy which has been improved more slowly by the labours of his followers. We now propose to treat of it at some length, endeavouring to lay before our readers as complete a view of this part of science as the nature of our work will permit.

We begin with laying down some definitions, and demonstrating some properties, of elliptical spheroids.

Def. 1. A solid generated by the revolving of an ellipse about either axis is called a spheroid of revolution. If the ellipse revolve about the less axis, the spheroid is oblate; if about the greater axis, it is oblong.

Let \( k \) and \( k' \) denote the two axes of the spheroid, \( k \) being that of revolution; and let \( x \) and \( y \) be two co-ordinates of a point in the surface of the spheroid, having their origin in the centre, \( x \) being parallel to the axis of revolution, and \( y \) perpendicular to it; then the equation of the spheroid, whether oblate or oblong, will be

\[ \frac{x^2}{k^2} + \frac{y^2}{k'^2} = 1. \]

Def. 2. An elliptical spheroid, in general, or an ellipsoid, is a solid bounded by a finite surface of the second order. Let ACB and ADE (Plate CI. fig. 1: this figure represents one eighth of an ellipsoid contained in one of the solid angles formed by the three principal sections) be two ellipses that have the same axis AO, the same centre O, and their planes perpendicular to one another: from any point K in the common axis, let there be drawn ordinates in both ellipses, as KC and KD; then, having described an ellipse of which KC and KD are the semiaxes, the periphery DMC of that ellipse will be in the surface of the ellipsoid. This solid figure has a centre, three axes crossing one another at right angles in the centre, and three principal sections made by planes passing through every two of its axes.

Let \( k, k', k'' \) denote the three semiaxes, viz. \( k = OB, k' = OE, k'' = OA \); and let \( x, y, z \), denote three rectangular co-ordinates of a point M in the surface, the co-ordinates being parallel to the axes, and having their origin in the centre, viz. \( MN = x, NK = y, OK = z \); then the equation of the surface will be

\[ \frac{x^2}{k^2} + \frac{y^2}{k'^2} + \frac{z^2}{k''^2} = 1; \]

as it is easy to prove from the foregoing construction.

The ellipsoid becomes a sphere when all the three axes are equal: it becomes a spheroid of revolution when two of them are equal.

1. If any plane cut an elliptical spheroid, the section will be an ellipse. In the spheroid of revolution, a section made by a plane perpendicular to the axis of revolution is a circle. All this follows so easily from the nature of the solids, that we need not stop to give a formal demonstration.

2. If a straight line cut two concentric ellipses that are similar and similarly situated, the parts of it between the outer and inner peripheries are equal to one another.

Let AHBK and MDNC (Plate CI. fig. 2) be two similar and similarly situated ellipses that have the same centre O; and let the straight line AB cut them both; then AC and BD are equal. Bisect CD in L, and through L and the common centre draw the straight line HMNK to cut both ellipses. Because the ellipses are similar and similarly situated, and that CD is an ordinate of the diameter MN, it is plain that AB will be an ordinate of the diameter HK; wherefore, AB and CD being both bisected in L, AC is equal to BD. 3. If there be two ellipses, one within the other, such that any straight line being drawn to cut them, the parts of it between their peripheries are equal to one another, these ellipses are concentric, similar, and similarly situated.

Let D (fig. 2) be any point in the inner ellipse, and through D draw EF, terminating in the outer ellipse; then, if we make FG = DE, G must be a point in the inner ellipse. Hence all the points of the inner curve are determined when the outer ellipse and the point D are given: wherefore there cannot be two different curves, both passing through D, that will answer the conditions. But an ellipse described through D, concentric with the outer ellipse, and similar to it, and similarly situated, will answer the conditions (2). Wherefore the two ellipses are concentric, similar, and similarly situated.

4. If a straight line be drawn to cut two elliptical spheroids that have the same centre, and are similar and similarly situated, the part of it between the outer and inner surfaces will be equal to one another.

Conceive a plane, which contains the straight line, to pass through the common centre of the solids; the sections made by the plane will be concentric ellipses (1); and these will be similar and similarly situated, because the solids are so; wherefore the parts of the straight line between the surfaces are equal (2).

5. If two elliptical spheroids that have the same centre, and are similar and similarly situated, be cut by a plane, the two sections will be concentric ellipses that are similar and similarly situated.

For the sections are ellipses (1); and, any straight line being drawn to cut them, the parts of it between the peripheries will be equal (4). Wherefore the ellipses are concentric, similar, and similarly situated (3).

6. Let ADE and CFG (fig. 3) be two concentric ellipses that are similar and similarly situated; let AO and CO, in the same straight line, be two of their axes, and let DE, drawn through C, be perpendicular to AO; then if CF and CG be two chords of the interior ellipse that make equal angles with the axis CO, and if the chords DM and DN of the exterior ellipse be drawn respectively parallel to CF and CG; the sum of CF and CG will be equal to the sum or difference of DM and DN, according as they both fall on the same side, or on different sides of DE.

For draw EP parallel to CF, and it will likewise be parallel to DM. Because CF and CG are equally inclined to CO and to DE, it is plain that DN and EP, which are parallel to CF and CG, are likewise equally inclined to DE; consequently DN = EP. Draw a straight line through the common centre to bisect DM in L, and that straight line will likewise bisect EP, parallel to DM, in H: and because the ellipses are similar and similarly situated, the same straight line will likewise bisect the chord CF of the interior ellipse, in K. Because DC = CE, therefore DL + EH = 2CK = CF. Therefore DM + DN = 2DL + 2EH = 2CF = CF + CG.

The demonstration of the other case, when DM and DN fall on different sides of DE, is entirely similar.

Some general Properties, resulting from the Law of Attraction that obtains in Nature.

7. Let AB and EF (fig. 4) be two indefinitely slender pyramids, that are similar to one another, and both composed of the same homogeneous matter, which attracts in the inverse proportion of the square of the distance; the attractions of the pyramids upon particles placed at the vertices A and E are proportional to the length of the pyramids.

Conceive each of the pyramids to be divided into an indefinitely great number of thin slices of equal thickness, by planes parallel to its base; then, if CD and GH be any two of these slices, their attractions upon particles placed at A and E will be proportional to \(\frac{CD}{AC}\) and \(\frac{GH}{EG}\): Now, these are equal; for the solids CD and GH having the same thickness, they are proportional to the sections CM and GN, that is, to AC\(^2\) and EG\(^2\), because the pyramids are similar. Wherefore the attraction of any one of the slices in the pyramid AB, upon a particle placed at A, is equal to the attraction of any one of the slices in EF upon a particle placed at E. Consequently, the whole attraction of the pyramid AB is to the whole attraction of the pyramid EF as the number of slices in AB to the number of slices in EF, that is, as the length AB to the length EF.

Cor. 1. The attractions of any portions of the pyramid are as the lengths of the portions. For the attractions are proportional to the number of slices in the portions, that is, as the lengths.

Cor. 2. If the pyramids have different densities, their attractions are proportional to the lengths multiplied by the densities. For, in this case, the attraction of each slice will be proportional to its density; wherefore the attractions will be as the densities multiplied by the number of slices, or as the densities multiplied by the lengths.

8. If there be two similar solids composed of the same homogeneous matter, which attracts in the inverse proportion of the square of the distance; any two particles of matter, similarly situated with regard to the solids, will be attracted by them with forces that are proportional to any of the homologous lines of the solids.

Because the solids are similar, they may be resolved into an indefinitely great number of slender pyramids, and frustums of pyramids, that are similar to one another, and similarly placed in the solids; each pyramid having its vertex at one of the attracted particles. The direct attractions of any corresponding pair of pyramids will have constantly the same ratio to one another; for they will be as the lengths of the pyramids or frustums (7); that is, because the solids are similar, as any two homologous lines of the solids. Wherefore the whole attractive forces, compounded of all the direct attractions which act in directions that make the same angles with one another, will likewise have to one another the proportion of any two of the homologous lines of the solids.

Cor. If the two solids have different densities, their attractions will be proportional to the densities multiplied by any homologous lines of the solids (7, Cor. 2).

9. If there be two concentric elliptical spheroids that are similar and similarly situated, a particle placed anywhere within the inner surface will be in equilibrium, or will be urged equally in all opposite directions by the shell of homogeneous matter contained between the two surfaces, supposing the law of attraction to be that of the inverse proportion of the square of the distance.

Let P (Plate CI. fig. 5) be a particle placed within such a shell, and let a slender double pyramid, having P for the common vertex, be extended to meet the surfaces of the solid on both sides of P. The portions of the pyramid AGHB and CEFD between the surfaces on opposite sides of P, will have equal lengths (4); wherefore these portions will attract a particle placed at P with equal forces (7, Cor. 1). The same thing may be proved of all the pyramids which have their vertices at P, and fill the spheroids. Wherefore P is attracted equally in all opposite directions by the homogeneous matter contained between the surfaces of the spheroids.

10. To find the attractive force of an indefinitely slender prism, acting in a direction parallel to the prism, upon a particle of matter placed anywhere.

Let BC (Plate CI. fig. 6) be a prism of homogeneous matter, upon the indefinitely slender base CH, and Attraction. Let a particle be placed at A; draw AB and AC to the extremities of the prism, and AE to any point in it; and draw AD perpendicular to BC. Let S = base CH, and put AD = a, DE = x; the element of the prism is \( S \times dx \), the element of the attraction in the direction AE is \( \frac{S \times dx}{AE^2} \), and the element of the attraction in the direction parallel to the prism is \( \frac{S \times dx}{AE^2} \times \frac{DE}{AE} = \frac{S \times dx}{(a^2 + x^2)^{3/2}} \).

Now, \( \int \frac{S \times dx}{(a^2 + x^2)^{3/2}} = \text{Const.} - \frac{S}{\sqrt{a^2 + x^2}} = \text{Const.} - \frac{S}{AE} \);

and the constant quantity is determined by making the fluent begin at the end of the prism nearer to A; wherefore the whole attractive force of the prism, in the direction parallel to the prism, is

\[ S \times \left( \frac{1}{AB} - \frac{1}{AC} \right). \]

Cor. In like manner may the attractive force of the prism be found, when the attraction of the particles is proportional to any function of the distance.

Let \( AB = f, AC = f' \); suppose that \( \varphi(f') \) is the function of the distance that expresses the law of attraction; and put \( \int df' \cdot \varphi(f') = Y(f') \); then the attraction parallel to the prism is

\[ S \times \left[ Y(f') - Y(f) \right], \]

observing that the attraction is always positive.

Attraction of spheres.

11. Spheres of the same homogeneous matter attract particles placed on their surfaces with forces proportional to their radii.

Spheres being similar solid figures, this proposition is no more than a particular case of what was before proved (8).

Cor. If the spheres have different densities, the attractions at their surfaces are proportional to their radii multiplied by their densities. (5, Cor.)

12. The force with which a particle, placed anywhere within a sphere of homogeneous matter, is urged towards the centre, is proportional to its distance from the centre.

Conceive a concentric sphere to be described, which contains the attracted particle in its surface; the matter between the two surfaces will exert no force on the particle (9), which will therefore be urged to the centre, only by the attraction of the inner sphere, in the surface of which it is placed; but this force is proportional to the radius of the sphere, or to the distance of the particle from the centre (11).

13. Let PNQ and ABC (fig. 7) be two spheres of the same homogeneous matter, which attracts in the inverse proportion of the square of the distance; let the centres of the spheres be at M and D, and take MR equal to the radius of the sphere ABC, and ED equal to the radius of the sphere PNQ; the attractions of the spheres upon particles placed at R and E are to one another as the squares of the radii of the spheres.

In the spheres draw two great circles perpendicular to the diameters PQ and AC, that pass through the points R and E; and let PpQ and AaC be two great circles, making equal indefinitely small angles NMP and BD6 with the great circles PNQ and ABC. Let HK and FG, parallel to PQ and AC, be any two chords of the circles PpQ and AaC, that subtend similar arcs, or arcs containing the same number of degrees; and through HK and FG, let planes perpendicular to the circles PpQ and AaC, be drawn to cut the portions of the spheres contained in the angles NMP and BD6; join RH, RK, MH, MK, DF, DG, EF, EG. Because the arcs subtended by HK and FG are like parts of their circumferences, it is plain that the angle RMH = EDF, and RMK = EDG. And because ED = MH = MK, and RM = DF = DG (hyp.), therefore RH = EF, and RK = EG.

Conceive the chords HK and FG, together with the planes passing through them, to change their place a little, so as to describe two slender prisms, or elements of the portions of the spheres contained in the angles NMP and BD6. It is plain that MX and DO, the distances of the chords HK and FG from the centres of their circles, are constantly proportional to MN and DB, the radii of the spheres; wherefore XT and OS, the perpendicular sections of the small prisms, are similar figures, and have to one another the same ratio that MX² has to DO², or MN² to DB². Now, the attraction of the prism HK urging a particle at R to the centre M, is

\[ XT \times \left( \frac{1}{RH} - \frac{1}{RK} \right), \]

and the attraction of the prism FG urging a particle at E to the centre D, is

\[ OS \times \left( \frac{1}{EF} - \frac{1}{EG} \right). \]

But in consequence of what was proved,

\[ \frac{1}{RH} - \frac{1}{RK} = \frac{1}{EF} - \frac{1}{EG}; \]

wherefore the attractions of the prisms are to one another as XT to OS, or as MN² to DB². The same thing may be proved of all the elements of the two portions of the spheres contained in the angles NMP and BD6; wherefore those portions attract particles at R and E with forces proportional to the squares of the radii of the spheres. But because the small angles NMP and BD6 are equal, each of the spheres may be divided into an equal number of such portions; wherefore the attractions of the whole spheres upon particles placed at R and E are proportional to the squares of the radii of the spheres.

Cor. This proposition is true when the particles of matter attract one another with forces proportional to any proposed function of the distance.

Let RH = EF = f, and RK = EG = f'; then, adopting the same notation as before (10, Cor.), the attractions of the prisms urging particles placed at R and E to the centres M and D, are respectively XT × [Y(f') − Y(f)], and OS × [Y(f') − Y(f)]; consequently, those attractions have the same proportion that XT has to SO, or MN² to DB². Wherefore the attractions of the whole spheres are in the same proportion.

14. A particle placed anywhere without a sphere of homogeneous matter which attracts in the inverse proportion of the square of the distance, will be urged to the centre of the sphere with a force that is inversely proportional to the square of the particle's distance from the centre.

Let ABC (Plate CL fig. 8) be the sphere, O its centre, and P a particle without the sphere; conceive a concentric sphere PMN, of the same homogeneous matter with the sphere ABC, to be described with the radius PO. Then, by the last proposition, the attraction of the sphere ABC upon the particle P is to the attraction of the sphere PMN, upon a particle placed at A, as AO² to PO². But the attraction of the sphere PMN upon a particle placed at A is equal to the attraction of the sphere ABC upon the same particle; for the attraction of the matter between the two spherical surfaces exerts no force upon a particle at A (9). Therefore, in the proportion set down above, the two middle terms are con- Attraction stantly the same wherever the point P is placed without the sphere ABC; consequently the first term of the proportion must follow the inverse ratio of the last term; that is, the attraction of the sphere ABC upon the external particle at P is inversely proportional to PO².

15. The same law of attraction being supposed, a homogeneous sphere will attract a particle placed without it, with the same force as if all the matter of the sphere were collected in the centre.

Let f denote the distance of the particle from the centre; then it follows, from the last proposition, that the attraction of the sphere upon the particle will have for its measure \( \frac{A}{f^2} \); A denoting a constant quantity that will be determined by any particular case; that is, by the actual attractive force corresponding to any determinate distance from the centre. Let r denote the radius of the sphere, and M its mass; then no part of the matter of the sphere being nearer the attracted particle than \((f - r)\), and none of it more remote than \((f + r)\), the attraction of the sphere on the particle will be greater than \( \frac{M}{(f + r)^2} \) and less than \( \frac{M}{(f - r)^2} \). Therefore \( \frac{A}{f^2} \) is always contained between those limits, which requires that \( A = M \). For, if A were greater than M, such values of f might be found as would make \( \frac{A}{f^2} \) equal to or greater than \( \frac{M}{(f - r)^2} \); and if A were less than M, such values of f might be found as would make \( \frac{A}{f^2} \) equal to or less than \( \frac{M}{(f + r)^2} \). Therefore \( A = M \); and the attraction of the sphere is equal to \( \frac{M}{f^2} \), or the same as if all the matter were collected in the centre.

If the radius of the sphere \( = r \), the density of the matter contained in it \( = d \); then the mass, or M, \( = \frac{4 \pi r^3 d}{3} \) (\( \pi \) being the circumference of the circle whose diameter is unit), and the attraction of the sphere at the distance \( f \) from the centre \( = \frac{4 \pi r^3 d}{3 f^2} \). This is still true at the surface of the sphere when \( f = r \), so that the attraction at the surface \( = \frac{4 \pi r^3 d}{3} \); which expression, with the help of what is proved in (12), enables us to compare the intensities of the attractions of homogeneous spheres, at all distances from the centre, without or within the surfaces.

Cor. 1. A shell of homogeneous matter contained between two concentric spherical surfaces, will attract a particle placed without it, with the same force as if all the matter of the shell were collected in its centre.

For the attractive force of such a shell is equal to the difference of the attractions of two concentric spheres of the same homogeneous matter with the shell.

Cor. 2. A sphere composed of concentric shells, that vary in their densities according to any law, will attract a particle placed without it, with the same force as if all the matter were collected in the centre.

For this having been proved of one shell (Cor. 1), it must be true of any number of shells.

If \( \varphi(r) \) denote the density at the distance \( r \) from the centre, the quantity of matter in the sphere will be \( = \frac{4 \pi}{f} \int \varphi(r) \cdot r^2 dr \); and the attraction on a particle without the sphere at the distance \( f \) from the centre \( = \frac{4 \pi}{f^2} \int \varphi(r) \cdot r^2 dr \).

16. Two spheres, each composed of concentric shells of variable density, attract one another with the same force as if all the matter of each were collected in its centre.

For the attraction of a sphere A upon every particle of another sphere B will remain the same, if we suppose all the matter of A to be collected in its centre (15). But the attraction of any particles of matter placed in A's centre upon the sphere B is equal and opposite to the attraction of B, upon the same matter so placed; and again, the attraction of B upon all the particles placed in the centre of A, will remain unchanged, if we suppose the matter of B to be collected in its centre. Wherefore A attracts B with the same force as if the matter of each were collected in its centre.

17. Supposing that the particles of matter attract with a force proportional to the distance, a body of any shape will attract a particle of matter placed anywhere with the same force, and in the same direction, as if all the matter of the body were collected in its centre of gravity.

Suppose that the attracted particle is placed at P (fig 9), and the centre of gravity of the attracting body at G; join PG, and let any plane pass through that line. Let L be a small part, or element of the body, and from L draw LK perpendicular to the plane passing through PG, and KF perpendicular to PG; join PL and PK. Put dm to denote the quantity of matter, or the mass of the element L; then its attractive force, urging the particle in the direction PL, is \( = PL \times dm \), which, by the resolution of forces, is equivalent to the two forces \( PK \times dm \) and \( KL \times dm \); and again, the single force \( PK \times dm \) is equivalent to the two forces \( FK \times dm \), and \( PF \times dm = PG \times dm + GF \times dm \). Wherefore, the attraction of the element L upon the particle at P is equivalent to these four separate forces, viz. \( PG \times dm, GF \times dm, PK \times dm, KL \times dm \), which urge the particle P respectively in the directions PG, GF, FK, KL. But, from the nature of the centre of gravity, the sum of all the forces, \( KL \times dm \), that urge the particle P to one side of the plane passing through PG, is just equal to the sum of the forces that urge it to the other side of the same plane; and the sum of all the forces, \( FK \times dm \), that urge P to one side of the line PG, is just equal to the sum of the forces that urge it to the other side of the same line; and the sum of all the forces, \( GF \times dm \), that urge P towards the point G, is just equal to the sum of the forces that urge it from the same point. Wherefore all the preceding forces mutually destroy one another, excepting the forces \( PG \times dm \), the sum of which, when extended to all the elements of the attracting body, is \( = PG \times \text{mass of the body} \). Wherefore the whole attraction upon P is the same as if all the matter of the body were collected in its centre of gravity.

Cor. Supposing that the particles of matter attract with a force proportional to the distance, a homogeneous sphere will attract a particle placed anywhere, in the same manner as if all the matter of the sphere were collected in the centre.

For the centre of gravity of a homogeneous sphere is the same as the centre of its figure. This corollary is likewise true of a sphere composed of concentric shells of variable density; and it is easy to apply the demonstration of (16) to prove that, in this law of attraction, two spheres, each composed of concentric shells of variable density, will attract one another with the same force as if the matter of each were collected in its centre.

18. To investigate what are the laws of attraction in re- Attraction.

Attraction, gard to the distance according to which a shell of homogeneous matter, contained between two concentric spherical surfaces, will attract a particle placed without it, in the same manner as if all the matter of the shell were collected in the centre.

It has been proved that this property actually belongs to homogeneous shells in the law of attraction which obtains in nature, and likewise when the particles of matter attract with a force proportional to the distance; but it is interesting to know whether it is confined to these two cases alone, or extends to other laws of attraction. This can only be discovered by a direct analysis.

Let \( r = PC \) (fig. 10), the distance of the attracted point from the centre of the shell; \( u = CA \), the radius of the inner surface of the shell; \( f = PM \), the distance of P from any point in the surface. Having drawn the diameter AD through P, let AMD and AND be two great circles, making with one another an indefinitely small angle MAN = \( dq \); and let two small circles BMG, bmg, indefinitely near one another, of which A and D are the poles, meet the former circles in M, N, m, n; and draw MS, NS to the centre of the circle BMG. Put \( \theta \) for the measure of the arc AM; then \( MS = u \sin \theta \); \( MN = dq \cdot u \sin \theta \); \( MM = u^2 \); and the quadrilateral space \( MNam = u^2 dq \cdot dq \cdot db \cdot \sin \theta \). We may suppose the thickness of the shell indefinitely small, since, if the property belong to an elementary shell indefinitely thin, it will be true of one of a determinate thickness, which can be resolved into such elements. Suppose the thickness of the shell to be \( du \); then the quantity of matter in the part standing upon the quadrilateral space \( MNam = u^2 du \cdot dq \cdot dq \cdot db \cdot \sin \theta \). Let \( \varphi(f) \) represent the direct attraction of a particle at M in the direction PM; then its attraction directed to the centre C = \( \varphi(f) \times \frac{PS}{PM} = \frac{r-u \cos \theta}{f} \times \varphi(f) \); and the attraction of the element of the shell in the same direction = \( u^2 du \cdot dq \cdot dq \cdot db \cdot \sin \theta \times \frac{r-u \cos \theta}{f} \cdot \varphi(f) \). This expression is proportional to \( dq \), when \( \theta \) and \( f \) remain constant; and therefore (denoting by \( \pi \) the circumference of the circle whose diameter is unit) the attraction of the whole zone contained between the small circles BMG, bmg, will be

\[ 2\pi \cdot u^2 du \cdot dq \cdot db \cdot \sin \theta \cdot \frac{r-u \cos \theta}{f} \cdot \varphi(f) \]; and the attractive force of the whole shell will be

\[ 2\pi \cdot u^2 du \int dq \cdot db \cdot \sin \theta \cdot \frac{r-u \cos \theta}{f} \cdot \varphi(f) \];

the fluent to be extended from \( \theta = 0 \) to \( \theta = \pi \).

Again, the quantity of matter in the shell is \( = 4\pi \cdot u^2 du \); and the attraction of this matter placed in the centre, at the distance \( r \) from P, is \( = 4\pi \cdot u^2 du \cdot \varphi(r) \).

If now we equate the attraction of the shell to the attraction of its matter placed in the centre, and leave out the factors common to both, we shall get

\[ 2\varphi(r) = \int dq \cdot db \cdot \sin \theta \cdot \frac{r-u \cos \theta}{f} \cdot \varphi(f) \],

the limits of the integral being the same as before.

But \( f^2 = r^2 - 2ru \cos \theta + u^2 \); then \( dq \cdot db \cdot \sin \theta = \frac{f df}{ru} \); also \( r-u \cos \theta = \frac{f^2 + r^2 - u^2}{2r} \); wherefore, by substitution, we get

\[ 4\varphi(r) \cdot u = \int (f^2 + r^2 - u^2) \cdot df \cdot \varphi(f) \];

or, which is equivalent, \( 4\varphi(r) \cdot u = \int (f^2 + r^2 - u^2) \cdot df \cdot \varphi(f) \);

the limits of this integral being from \( f = r-u \) to \( f = r+u \), which correspond to \( \theta = 0 \) and \( \theta = \pi \).

Now let \( \int df \cdot \varphi(f) = \Psi(f) \); and \( \int df \cdot \int df \cdot \varphi(f) = \text{Attraction} \).

\[ \int df \cdot \Psi(f) = \Psi(f) \]; then, by taking the fluents between the proper limits, we get

\[ 4\varphi(r) \cdot u = 2r \left[ \Psi(r+u) - \Psi(r-u) \right] - 2 \left[ \Psi(r+u) - \Psi(r-u) \right] \].

If we develop the binomial functions in the last expression, all the even powers of \( u \) will disappear, and the odd powers only will remain; these last terms being all contained in this general formula,

\[ \frac{4}{1,2,3,...2n+1} \left\{ r \cdot \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} - \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} \right\} \cdot u^{2n+1} \]

and, observing that \( \frac{d\Psi(r)}{dr} = r\Psi(r) \), the same expression will become

\[ \frac{4}{1,2,3,...2n+1} \left\{ r \cdot \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} - \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} \right\} \cdot u^{2n+1} \]

which, again, is more simply expressed thus, viz.

\[ \frac{4}{1,2,3,...2n+1} \left\{ r \cdot \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} - \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} \right\} \cdot u^{2n+1} \]

Wherefore, by substituting the development instead of the functions, and then dividing by \( 4r^2u \), we get

\[ \varphi(r) = \frac{d\Psi(r)}{dr} + \frac{1}{1,2,3,...2n+1} \left\{ r \cdot \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} - \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} \right\} \cdot u^{2n+1} \]

From the nature of the function \( \Psi(r) \), we get \( \varphi(r) = \frac{d\Psi(r)}{dr} \); wherefore each of the remaining terms must be separately equal to nothing; hence

\[ d \left\{ r \cdot \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} - \frac{d^{2n+1}\Psi(r)}{dr^{2n+1}} \right\} = 0 \];

from which we find \( r\Psi(r) = \frac{1}{A}Ar^2 + \Lambda r - A^* \), \( A, A', A'' \) being arbitrary constant quantities; and this value of \( r\Psi(r) \), it is plain, will likewise render all the succeeding terms of the development evanescent. Therefore

\[ \varphi(r) = \frac{d\Psi(r)}{dr} = Ar + \frac{A^*}{r^2} \].

Thus the most general expression of the law of attraction, that possesses the property in question, is a combination of the two laws above mentioned, with each of which it coincides, according as we make the one or other of the constant quantities equal to nothing. We have, therefore, a direct proof that the law of nature is the only one which will make the attraction decrease as the distance increases, and in which a spherical shell, or a sphere, will attract in the same manner as if all the matter were collected in the centre.

Laplace has arrived at the same conclusion by a different process. (Mec. Celeste, livre ii, chap. 2, No. 12, Rem. part.)

Attraction of Spheroids of Revolution.

19. Let APBQ (Plate CII. fig. 11) and CMDN be two concentric ellipses, similar to one another, and similarly rods of situated, of which AB and CD are either the greater or less revolution. Attraction axes; and let PCQ be perpendicular to AB. Conceive the ellipses to revolve about PQ, so as to describe an indefinitely small angle; then, supposing the law of attraction to be inversely proportional to the square of the distance, the thin solid of homogeneous matter described by the ellipse APBQ will attract a particle placed at P, in a direction perpendicular to any plane passing through PQ, with the same force that the thin solid of the same matter described by the ellipse CMDN, will attract a particle placed at C perpendicularly to the same plane.

From C draw CM, CN, making equal angles with CD, and PR, PT respectively parallel to CM, CN; and let CM, CN, PR, PT be drawn in the same manner, and indefinitely near the former lines. While the ellipses revolve about PQ, the small sectors will describe pyramids that have their vertices at C and P. It is manifest that the pyramids so described are similar; for their angles at C and P in the planes of the ellipses are equal; and their other angles, described by revolving about PQ, are likewise equal, because the sectors are equally inclined to that axis. Therefore the direct attractions of all the small pyramids upon the particles P and C are proportional to the lengths PR, PT, CM, CN (7); and consequently the forces that urge the particles P and C in a direction at right angles to any plane passing through PQ are proportional to the perpendiculars let fall upon that plane from R, T, M, N. But because PR, PT, CM, CN are equally inclined to PQ, they will make equal angles with any plane passing through PQ; wherefore the perpendiculars drawn to the plane from R, T, M, N, will be respectively proportional to PR, PT, CM, CN. But CM + CN = PR + PT (6); wherefore the sum of the perpendiculars drawn to the plane from M and N will be equal to the sum of the perpendiculars drawn to it from R and T. Consequently the force of the pyramids PR and PT, which urges the particle P at right angles to the plane, is equal to the force of the pyramids CM and CN, which urges the particle C in a parallel direction. The same thing is true of all the small pyramids that make up the thin solids described by the ellipses APBQ and CMDN; and it is therefore true of the whole solids.

It is to be observed that when the pyramids PR and PT fall on opposite sides of PQ, it is the difference of their attractions which is equal to the sum of the attractions of CM and CN; and it is the difference of the perpendiculars let fall from T and R on opposite sides of the plane, which is equal to the sum of the perpendiculars let fall from M and N.

20. Let APBQ be a spheroid of revolution, PQ the axis of revolution, and ACB a plane through the centre perpendicular to PQ (Plate CII. fig. 12). If D be a particle in the surface of the spheroid, and DL perpendicular to the plane ACB; then the attraction of a spheroid on a particle placed at the pole P, will be to the force with which a particle placed at D is attracted in the direction DL, as PC is to DL.

Through D draw a plane parallel to the plane ACB, and let the plane so drawn cut the axis PQ in F; draw the straight line DFE to terminate in the spheroid, and describe another spheroid through F, having the same centre with the spheroid APBQ, and similar to it, and similarly situated. Conceive an indefinitely great number of planes, making indefinitely small angles with one another, to be drawn through DE, so as to divide the two spheroids into an indefinitely great number of thin solids or slices; then the sections which every one of the planes make with the spheroids will be similar ellipses, having the same centre (5); and it is manifest that a straight line drawn through F at right angles to DE, in any one of the planes, will pass through the centre of the two ellipses contained in it, and will coincide with an axis of Attraction each. Wherefore the force with which every one of the slices or elements of the spheroid APBQ attracts a particle placed at D in the direction DL, is equal to the force with which the corresponding slice or element of the spheroid GFKH attracts a particle placed at F in the direction FC (19). Wherefore, the whole attraction of the spheroid APBQ upon a particle at D, in the direction DL, is equal to the whole attraction of the spheroid GFKH upon a particle at F. But the attractions of the spheroids APBQ and GFKH upon particles placed at P and F, are to one another as PC to FC (8). Wherefore, the attraction of the spheroid APBQ upon a particle at P, is to the force with which the same spheroid attracts a particle at D, in the direction DL, as PC is to FC or DL.

21. Let APBQ be a spheroid of revolution, and PQ the axis of revolution as before. If D be a particle in the surface ADPQ (fig. 13) a section through D, and the axis PQ and DL perpendicular to PQ; the attraction of the spheroid upon a particle at A will be to the force with which a particle at D is attracted, in the direction DL, as AC is to DL.

Through D draw a plane perpendicular to AB, which cuts the section ADB in the straight line DEF; and let a spheroid FGHK be described through F, having the same centre with the spheroid APBQ, and similar to it, and similarly situated. Then, conceiving the two spheroids to be divided into an indefinitely great number of such slices by planes passing through DE, the force with which every slice or element of the spheroid APBQ attracts a particle at D in the direction DL, will be equal to the force with which the corresponding slice or element of the spheroid FGHK attracts a particle at F (19). But the attractions of the spheroids APBQ and FGHK upon particles placed at A and F are to one another as AC to CF (8). Wherefore, the attraction of the spheroid APBQ upon a particle at A is to the force with which the same spheroid attracts a particle at D, in the direction DL, as AC to FC or DL.

The two last propositions will enable us to find both the direction and the intensity of the attraction of a homogeneous spheroid of revolution upon a particle placed anywhere on the surface, when we have ascertained the attractive forces at the poles and at the circumference of the circular section made by a plane through the centre perpendicular to the axis. For the whole attraction at any point is the compound force arising from the attractions perpendicular to the axis, and parallel to it. The next object of our research is, therefore, to determine the two forces above mentioned, viz. the attraction at the poles, and at the circular section equally distant from both poles.

22. Let ABD be an indefinitely slender pyramid, of which the base BD is perpendicular to the edge AD (fig. 14); let B = base BD, and f = length AD; then \( \frac{B}{f} \) is the attraction of the whole matter of the pyramid upon a particle placed at the vertex A.

Let AM = x, then the section MN parallel to the base BD = \( \frac{Bx^2}{f^2} \), and MP = element of the prism = \( \frac{Bx^2dx}{f^2} \), and the attraction of the element upon a particle placed at A = \( \frac{MP}{AM^2} = \frac{Bdx}{f^2} \), the fluent of which is \( \frac{Bx}{f^2} \) = attraction of the pyramid AM upon a particle at A. And, when \( x = f \), this becomes \( \frac{B}{f} \) = attraction of the pyramid AD upon a particle placed at A. Attraction.

23. To investigate the attraction of a homogeneous spheroid of revolution upon a particle placed at the pole.

Let \( P \) (fig. 15) be the pole, \( PCQ \) the axis of revolution, and \( APBQ \) a section of the spheroid by a plane passing through \( PQ \), and any point \( M \), in the surface; draw \( PM \), \( PM \) indefinitely near \( PM \), and \( MM \) perpendicular to \( PM \). Conceive the plane \( PMQ \) to revolve about \( PQ \), so as to describe the indefinitely small angle \( BCO \); then the small triangle \( MPm \) will describe a slender pyramid, having its vertex at \( P \), and of which the base is a rectangle, contained by \( MM \) and \( RT \); for the point \( M \) moving parallel to \( R \), it will describe a line equal and parallel to that described by \( R \), namely, to \( RT \).

Let \( PM = f \), and the angle \( KPM \), which \( PM \) makes with a perpendicular to the axis, \( = \delta \), and the indefinitely small angle \( BCO = d\phi \). Then \( MM = f d\phi \), \( RT = CR \times d\phi = f \cos \delta \times d\phi \), and \( B \), the base of the slender pyramid described by the triangle \( MPm = d\phi \times d\phi \cos \delta \times f^2 \); therefore the direct attraction of the pyramid on a particle at \( P = \frac{B}{f} \) (22) \( = d\phi \times d\phi \cos \delta \times f \), and the elementary attraction of the spheroid in the direction \( PC = \) direct attraction of the pyramid \( \times \frac{PS}{PM} = d\phi \times d\phi \cos \delta \times \sin \delta \times f \).

Again, let \( MR = x \), \( CR = y \), \( PC = k \), \( AC = k' \); then \( y = f \cos \delta \), \( x = k - f \sin \delta \); if we substitute these values in the equation of the solid (Def. 1) we get

\[ \left( \frac{k-f \sin \delta}{k^2} \right)^2 + \frac{f^2 \cos^2 \delta}{k^2} = 1; \]

whence

\[ f = \frac{2k^2}{k^2} \cdot \frac{\sin \delta}{\cos^2 \delta + \frac{k^2}{k^2} \sin^2 \delta}. \]

By substituting the value of \( f \) just found in the preceding expression of the elementary attraction of the spheroid, it will become

\[ \frac{2k^2}{k^2} \cdot d\phi \cdot \frac{d\phi \cos \delta \cdot \sin^2 \delta}{\cos^2 \delta + \frac{k^2}{k^2} \sin^2 \delta}; \]

which must be integrated from \( \phi = 0 \) to \( \phi = 2\pi \), and from \( \delta = 0 \) to \( \delta = \frac{\pi}{2} \) denoting always the half-circumference when radius is unit.

In an oblate spheroid, \( k \) is less than \( k' \); put \( k^2 - k'^2 = k^2 \cdot e^2 \), and \( z = \sin \delta \); then the element of the attractive force will become, by substitution,

\[ \frac{2k^2}{k^2} \cdot d\phi \cdot \frac{z^2 dz}{1 + e^2 z^2} = \frac{2k^2}{k^2} \cdot d\phi \cdot \left\{ \frac{edz}{1 + e^2 z^2} \right\}; \]

and by integrating from \( z = 0 \) to \( z = 1 \), we get

\[ \frac{2k^2}{k^2} \cdot [e - \text{arc. tan } e] \]

for the force with which the matter between the planes \( PBQ \) and \( POQ \) urges the particle \( P \) to the centre. Therefore the whole attractive force of the spheroid upon a particle at \( P \) is

\[ \frac{4\pi}{3} \cdot \frac{k^2}{k^2} \cdot [e - \text{arc. tan } e]. \]

And because \( \frac{4\pi}{3} \cdot \frac{k^2}{k^2} = \text{mass of the spheroid } = M \), we get the measure of the attraction of the oblate spheroid upon a particle placed at the pole equal to

\[ \frac{3M}{k^2} \cdot [e - \text{arc. tan } e]. \]

In an oblong spheroid, \( k \) is greater than \( k' \); put \( k^2 - k'^2 = k^2 \cdot e^2 \), then the element of the attractive force will become, by substitution,

\[ \frac{2k^2}{k^2} \cdot d\phi \cdot \frac{z^2 dz}{1 + e^2 z^2} = \frac{2k^2}{k^2} \cdot d\phi \cdot \left\{ \frac{edz}{1 + e^2 z^2} \right\}; \]

whence, by proceeding as before, we get the measure of the attractive force of the oblong spheroid on a particle placed at the pole equal to

\[ \frac{3M}{k^2} \cdot \left\{ \frac{1}{2} \text{ hyp. log. } \frac{1 + e}{1 - e} \right\} \]

Cor. In an oblate spheroid differing little from a sphere, \( e^2 \) will be a very small fraction, of which we may reject the higher powers. When this is done, the preceding expression of the polar attraction, viz.

\[ \frac{4\pi}{3} \cdot \frac{k^2}{k^2} \cdot (e - \text{arc. tan } e), \]

will be \( 4\pi k \cdot (1 + e^2) \cdot \left( \frac{1}{2} - \frac{1}{2} e^2 \right) = \frac{4\pi k}{3} \cdot \left( 1 + \frac{2}{5} e^2 \right). \]

And, if \( R = k + r = k \sqrt{1 + e^2} \) be the radius of the equator, then \( \frac{2r}{k} = e^2 \), so that the attraction at the pole will be

\[ \frac{4\pi k}{3} \cdot \left( 1 + \frac{4}{5} \cdot \frac{r}{k} \right). \]

24. To investigate the attraction of a homogeneous spheroid of revolution on a particle placed in the circumference of the circular section made by a plane through the centre, at right angles to the axis of revolution.

Let \( P \) (Plate CII. fig. 16) be the pole, \( PC \) the axis of revolution, \( A \) a point in the circular section \( AOB \), made by a plane through the centre perpendicular to \( PC \). Let \( M \) be any point in the surface of the spheroid, \( AMO \) a section through \( A \) and \( M \) by a plane perpendicular to \( AOB \), \( AM \) a line in that plane indefinitely near \( AM \), and \( MM \) perpendicular to \( AM \), \( MR \) perpendicular to \( AO \), and \( RS \) to \( AR \). Conceive the plane \( AMO \) to revolve about \( A \), so as to describe an indefinitely small angle \( OAQ \); then the triangle \( AMm \) will describe a slender pyramid, having its vertex at \( A \), and of which the base is equal to a rectangle contained by \( MM \) and \( RT \); for the point \( M \) moving parallel to the point \( R \), it will describe a line equal to that described by \( R \), namely, to \( RT \).

Let \( AM = f \), the angle \( MAR = \delta \), and the angle \( OAQ = d\phi \); then \( MM = f d\phi \), and \( TR = AR \times d\phi = f \cos \delta \times d\phi \). Wherefore \( B = \) base of the pyramid described by \( AMm = d\phi \times d\phi \cos \delta \times f^2 \), and the direct attraction of the pyramid in the direction \( AM = \frac{B}{f} \) (22) \( = d\phi \times d\phi \cos \delta \times f \). Wherefore the elementary attraction of the spheroid in the direction \( AC = \) direct attraction of the pyramid \( \times \frac{AR}{AM} \times \frac{AS}{AR} = d\phi \cos \delta \times d\phi \cos^2 \delta \times f \).

Again, let \( MR = x \), \( RS = y \), \( CS = z \), \( CP = k \), and \( AC = k' \); then (Def. 1)

\[ \frac{x^2}{k^2} + \frac{y^2}{k^2} + \frac{z^2}{k^2} = 1. \]

But \( x = f \sin \delta \), \( y = AR \cdot \sin \phi = f \cos \delta \cdot \sin \phi \), and \( z = k' - f \cos \delta \cdot \cos \phi \); wherefore, by substitution, we get

\[ \frac{f^2 \sin^2 \delta}{k^2} + \frac{f^2 \cos^2 \delta \sin^2 \phi}{k^2} + \frac{(k' - f \cos \delta \cdot \cos \phi)^2}{k^2} = 1. \]

From this equation we get

\[ f = \frac{2k'}{\cos \delta \cdot \cos \phi}. \]

Let this value of \( f \) be substituted in the expression of the elementary attraction of the spheroid before found, and it will become Attraction.

\[2k' \cdot dp \cos^2 \phi \cdot \frac{d\theta \cos^2 \theta}{\cos^2 \theta + k^2 \sin^2 \theta}\]

which expression, when integrated from \( \phi = 0 \) to \( \phi = \pi \), and from \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \), will give the attraction of half the spheroid; and the double of it, viz.

\[4k' \cdot dp \cos^2 \phi \cdot \frac{d\theta \cos^2 \theta}{\cos^2 \theta + k^2 \sin^2 \theta}\]

being integrated between the same limits, will give the whole attraction of the spheroid.

In the oblate spheroid, \( k \) is less than \( k' \); let \( k^2 - k'^2 = k^2 e^2 \), and \( z = \sin \theta \); and, by substitution, the element of the attractive force will become

\[4k' \cdot dp \cos^2 \phi \cdot dz(1 - z^2) = \frac{4k' \cdot dp \cos^2 \phi}{1 + e^2} \left\{ \frac{e}{1 + e^2} \right\}\]

And by integrating from \( z = 0 \) to \( z = 1 \), we get

\[4k' \cdot dp \cos^2 \phi \cdot \frac{1 + e^2}{e^2} \left\{ \text{arc. tan. } e - \frac{e}{1 + e^2} \right\}\]

for the force with which the matter between the sections that contain the angle \( OAP \) attracts the particle \( A \) to the centre. But

\[\int dp \cos^2 \phi = \int \frac{dp}{2} (1 + \cos 2\phi)\]

\[= \frac{\pi}{2} + \frac{1}{4} \sin 2\phi\]

the value of which, between the limits \( \phi = 0 \) and \( \phi = \pi \), is \( \frac{\pi}{2} \); wherefore the attraction of the spheroid on a particle at \( A \) is equal to

\[2\pi \cdot k \cdot \frac{1 + e^2}{e^2} \left\{ \text{arc. tan. } e - \frac{e}{1 + e^2} \right\}\]

Because

\[\frac{4\pi \cdot k^2}{3} = \frac{4\pi \cdot k^2 \cdot (1 + e^2)}{3} = M\]

we get

\[2\pi \cdot (1 + e^2) = \frac{3M}{2k^2}\]

whence, by proceeding as before, we get the measure of the attractive force of the oblong spheroid upon a particle placed anywhere in the circumference of the circular section made by a plane through the centre at right angles to the axis, equal to

\[k \cdot \frac{3M}{2k^2 \cdot e^2} \left\{ \text{arc. tan. } e - \frac{e}{1 + e^2} \right\}\]

In the oblong spheroid, \( k \) is greater than \( k' \); put \( k^2 - k'^2 = k^2 e^2 \), then the element of the attractive force will become, by substitution,

\[4k' \cdot dp \cos^2 \phi \cdot \frac{dz(1 - z^2)}{1 - e^2} = 4k' \cdot dp \cos^2 \phi \cdot \frac{1 - e^2}{e^2} \left\{ \frac{e}{1 - e^2} - \frac{e}{1 - e^2} \right\}\]

whence, by proceeding as before, we get the measure of the attractive force of the oblong spheroid upon a particle placed anywhere in the circumference of the circular section made by a plane through the centre at right angles to the axis, equal to

\[k' \cdot \frac{3M}{2k^2 \cdot e^2} \left\{ \frac{e}{1 - e^2} - \frac{1}{2} \text{ hyp. log. } \frac{1 + e}{1 - e} \right\}\]

Cor. In an oblate spheroid, differing little from a sphere, the higher powers of \( e^2 \) may be neglected. The expression of the attractive force at the equator, viz.

\[2\pi \cdot k' \cdot \frac{1 + e^2}{e^2} \left\{ \text{arc. tan. } e - \frac{e}{1 + e^2} \right\}\]

will then become

\[2\pi \cdot k' \cdot (1 + e^2) \left( \frac{2}{3} - \frac{4}{5} e^2 \right) = \frac{4\pi k'}{3} \cdot (1 - \frac{1}{5} e^2)\]

And if \( k' \), the radius of the equator, \( = k + r \); then

\[\frac{2r}{k} = \frac{e}{k'}\]

(23, Cor.) and the attraction at the equator will be equal to

\[\frac{4\pi k'}{3} \left( 1 + \frac{e}{k'} \right) \left( 1 - \frac{2}{5} \cdot \frac{r}{k} \right) = \frac{4\pi k'}{3} \left( 1 + \frac{3}{5} \cdot \frac{r}{k} \right)\]

25. An oblate spheroid of revolution being given, it is required to find the measures of the attractive forces that urge a particle placed anywhere in the surface, in a direction perpendicular to the axis, and in a direction parallel to it.

Let \( k \) and \( k' \) be the semiaxes of the ellipse, by the revolution of which the spheroid is described, \( k \) being the axis about which it revolves; and let \( b \) be the perpendicular distance of the particle from the axis, and \( a \) its distance from the plane, drawn through the centre at right angles to the axis: then, from what was proved in (20) and (21), the attractions sought will be found by multiplying the attractions at the pole, and at the circular section equally distant from both poles, by \( \frac{a}{k} \) and \( \frac{b}{k'} \).

Thus we get the attraction in the direction of \( a \) equal to

\[a \times \frac{3M}{k \cdot e^2} \left\{ \text{arc. tan. } e - \frac{e}{1 + e^2} \right\}\]

and the attraction in the direction of \( b \) equal to

\[b \times \frac{3M}{2k^2 \cdot e^2} \left\{ \text{arc. tan. } e - \frac{e}{1 + e^2} \right\}\]

The same formulæ likewise serve for finding the attractions upon a particle placed anywhere within the spheroid. For the attraction upon a particle within the spheroid is equal to the attraction of a similar concentric spheroid, which contains the particle in its surface (9); and it is evident that the co-efficients which multiply \( a \) and \( b \) in the above expressions depend only upon the proportion of \( k \) and \( k' \), and they are therefore the same for all similar spheroids.

If we denote by \( A \) and \( B \) the co-efficients of \( a \) and \( b \) in the expressions of the attractive force found above, the whole attraction of the spheroid, which is compounded of the forces \( a \cdot A \) and \( b \cdot B \), will be \( \sqrt{a^2 A^2 + b^2 B^2} \).

And if \( \Psi \) denote the angle which the direction of this force makes with \( a \), or with the axis of the spheroid; then

\[\tan \Psi = \frac{b \cdot B}{a \cdot A}\]

Cor. In the very same manner we may determine the attractions of an oblong spheroid of revolution, upon a point on the surface, or within the solid.

26. If \( k, k', k'' \), the semiaxes of a homogeneous ellipsoid, be related to \( h, h', h'' \), those of another ellipsoid of the same matter, so that \( k^2 - k'^2 = h^2 - h'^2 \) and \( k'^2 - k''^2 = h'^2 - h''^2 \), the attractions perpendicular to the planes of the principal sections, which the first ellipsoid (Plate CII. fig. 17) exerts upon a point determined by the co-ordinates \( h \sin m, h' \cos m \sin n, h'' \cos m \cos n \), respectively parallel to \( k, k', k'' \), will be to the attractions which the second ellipsoid exerts upon a point determined by the co-ordinates \( h \sin m, h' \cos m \sin n, h'' \cos m \cos n \), respectively parallel to \( k, k', k'' \), in the direct proportion of the areas of the principal sections to which the attractions are perpendicular.

This proposition is an extension to all elliptical spheroids of what was proved of the sphere in (18). It is here enumerated of the ellipsoid, because the demonstration is not more difficult for that solid than for spheroids of revolution.

Let \( ABDM \) be an ellipsoid, the semiaxes of which are \( BC = h, EC = k' \); and \( AC = k'' \); and \( abdm \) another ellipsoid, of which the semiaxes are \( bc = h, ce = h' \), and \( ac = k'' \); these quantities being so related that \( k^2 - k'^2 = h^2 - h'^2 \) and \( k'^2 - k''^2 = h'^2 - h''^2 \). Also, let \( G \) be a point about the ellipsoid \( ABDM \), so determined that \( GH \), parallel to \( BC, = h \sin m \); \( HK \), parallel to \( CE, = k' \cos m \). Attraction. sin. \( n \); and \( CK = k' \cos. m \cos. n \); and let \( g \) be a point about the ellipsoid \( abdm \), so determined that \( gh \), parallel to \( bc = k \sin. m \); \( hk \), parallel to \( ce = k' \cos. m \sin. n \); and \( ck = k' \cos. m \cos. n \). Then the force with which the ellipsoid \( ABDM \) attracts a particle placed at \( G \) in the direction \( GH \), will be to the force with which the ellipsoid \( abdm \) attracts a particle placed at \( g \) in the direction \( gh \), as the area of the section \( AEDM \) to the area of the section \( acdm \), or as \( k^2 k'^2 \) to \( k' k'' \).

Let \( RP = k \sin. \phi \); \( PO = k' \cos. \phi \sin. \psi \); and \( CO = k' \cos. \phi \cos. \psi \); which suppositions are allowable, because they satisfy the equation of the ellipsoid (Def. 1), whatever be the angles \( \phi \) and \( \psi \). Draw \( CP'M \) through the centre, and \( CN \) indefinitely near it; then \( CP = \cos. \phi \sqrt{k^2 \sin^2 \phi + k'^2 \cos^2 \psi} \); and when \( \cos. \phi = 1 \),

\[ CM = \sqrt{k'^2 \sin^2 \phi + k'^2 \cos^2 \psi}; \text{ therefore } \frac{CP}{CM} = \cos. \phi. \]

Let the angle \( DCM = Y \); then \( \tan. Y = \frac{PO}{CO} = \frac{k'}{k} \tan. \phi \);

and, by taking the fluxions, \( \frac{dY}{d\phi} = \frac{k'}{k} \cdot \frac{d\phi}{\cos^2 \phi} \); but

\[ \frac{1}{\cos^2 Y} = 1 + \tan^2 Y = \frac{k'^2 \sin^2 \phi + k'^2 \cos^2 \psi}{k'^2 \cos^2 \phi} = \frac{CM^2}{k'^2 \cos^2 \phi}; \]

wherefore \( dY \cdot CM^2 = \text{twice the sector } MCN = k' k'' \cdot d\phi \).

And, in like manner, in the other ellipsoid, if \( rP = h \sin. \delta \);

\[ PO = h' \cos. \delta \sin. \psi; \text{ and } CO = h' \cos. \delta \cos. \psi; \text{ then } \frac{CP}{CM} = \cos. \delta, \text{ and twice the sector } men = k' k'' \cdot d\phi. \]

It is plain, from what has been shown, that, when \( \phi \) varies and \( \psi \) remains constant in the expressions of the co-ordinates, the points \( P \) and \( p \) will move along \( CM \) and \( cm \), so that, in every position,

\[ \frac{PC}{MC} = \frac{pc}{mc}. \]

Let \( Q \) and \( q \) be indefinitely near \( P \) and \( p \); and through \( P \) and \( Q \) draw lines parallel to \( MN \), and through \( p \) and \( q \) draw lines parallel to \( mn \). Let \( S \) denote the quadrilateral contained between the parallels drawn through \( P \) and \( Q \), and \( S' \) that contained between the lines drawn through \( p \) and \( q \): Then

\[ \frac{S}{MCN} = \frac{QC^2 - PC^2}{MC^2}, \text{ and } \frac{S'}{MCN} = \frac{qc^2 - pc^2}{mc^2}; \text{ therefore, since } \frac{PC}{MC} = \frac{pc}{mc}, \text{ and } \frac{QC}{MC} = \frac{qc}{mc}, \text{ it is manifest that } \frac{S}{S'} = \frac{MCN}{mcn} = \frac{k' k''}{k' k''}. \]

Upon the quadrilaterals \( S \) and \( S' \) let upright prisms \( RS \) and \( rs \) be erected, and be prolonged to meet the surfaces of the spheroids: join \( GR, GS, gr, gs \). Then,

\[ GR^2 = (h \sin. m - k \sin. \phi)^2 + (h' \cos. m \sin. n - k' \cos. \phi \sin. \psi)^2 + (h' \cos. m \cos. n - k' \cos. \phi \cos. \psi)^2; \]

\[ gr^2 = (h \sin. m - k \sin. \phi)^2 + (h' \cos. m \sin. n - k' \cos. \phi \sin. \psi)^2 + (h' \cos. m \cos. n - k' \cos. \phi \cos. \psi)^2; \]

And, by expanding these expressions, we get

\[ GR^2 = [(k^2 + (k'^2 - k^2)) \cos^2 \phi \sin^2 \phi + (k^2 - k'^2) \cos^2 \phi \cos^2 \psi] \]

\[ + (k^2 + (k'^2 - k^2)) \cos^2 \phi \sin^2 \phi + (k^2 - k'^2) \cos^2 \phi \cos^2 \psi] \]

\[ - 2(hk \sin. m \sin. \phi + h' k' \cos. m \cos. \phi \sin. n \sin. \phi \]

\[ + h' k' \cos. m \cos. \phi \cos. n \cos. \phi)]; \]

and \( gr^2 = [(k^2 + (k'^2 - k^2)) \cos^2 \phi \sin^2 \phi + (k^2 - k'^2) \cos^2 \phi \cos^2 \psi] \)

\[ + (k^2 + (k'^2 - k^2)) \cos^2 \phi \sin^2 \phi + (k^2 - k'^2) \cos^2 \phi \cos^2 \psi] \)

\[ - 2(hk \sin. m \sin. \phi + h' k' \cos. m \cos. \phi \sin. n \sin. \phi \]

\[ + h' k' \cos. m \cos. \phi \cos. n \cos. \phi)]. \]

These expressions are equal, because \( k^2 - k'^2 = h^2 - h'^2 \), and \( k'^2 - k^2 = h'^2 - h^2 \); therefore \( RG = rg = f \). And, in like manner, it is shown, that \( GS = gs = f' \).

Now, the attraction of the prism \( RS \) urging a particle at \( G \) in the direction \( GH \), is equal to \( S \times \left( \frac{1}{f} - \frac{1}{f'} \right) \) (10);

and the attraction of the prism \( rs \) urging a particle at \( g \) in the direction \( gh \), is \( S' \times \left( \frac{1}{f} - \frac{1}{f'} \right) \); wherefore these attractions are to one another as \( S \) to \( S' \), or as \( k' k'' \) to \( k' k'' \).

The same thing may be proved of all the elementary prisms that make up the two portions of the ellipsoids contained between the planes \( BCM, BCN, \) and \( bcm, bcn \); wherefore those portions attract particles placed at \( G \) and \( g \), with forces proportional to \( k' k'' \) and \( k' k'' \). But the two ellipsoids may be divided into an equal number of such portions; wherefore the ellipsoids attract particles placed at \( G \) and \( g \), in the directions \( GH \) and \( gh \), with forces proportional to \( k' k'' \) and \( k' k'' \), or to the sections \( AMDE \) and \( amde \).

Cor. 1. This proposition is true when the law of attraction is expressed by any function of the distance. The demonstration is the same as in the corollary of (13).

Cor. 2. If the two ellipsoids be so placed that their centres, and the planes of their principal sections, shall coincide, the surface of the one will be entirely within the other. Also the point which one ellipsoid attracts will be in the surface of the other, as is plain from the expressions of the co-ordinates. And hence the attraction of one ellipsoid upon a point without the surface is made to depend upon the attraction of another ellipsoid upon a point within the surface.

Cor. 3. When the ellipsoids become spheroids of revolution, the two principal sections through the axis of revolution become equal, and will be represented by any two sections whatever passing through the axis at right angles to one another. But, in this case, the attractions of the spheroids on the points may be reduced to two, one acting perpendicular to the axis, and one parallel to it; and it is plain that these attractions will be to one another as the areas of the sections perpendicular to their directions.

27. To find the attraction of an oblate spheroid upon a particle placed without the surface.

Let \( k \) be the radius of the equator, and \( k \) the axis of revolution; and let \( a \) be the perpendicular distance of the point without the spheroid from the plane of the equator, and \( b \) its distance from the axis. In the first place, it is necessary to determine the semiaxis of another oblate spheroid that shall contain the given point in its surface, and such, that it shall have the same centre, and its equator in the same plane, as the given spheroid; and likewise the difference of the squares of its semiaxes equal to the difference of the squares of the semiaxes of the given spheroid. Let \( k' \) denote the radius of the equator, and \( h \) the semiaxis of the required spheroid; then, because the attracted point is to be in the surface of the solid, we have \( \frac{a^2}{k^2} + \frac{b^2}{k^2} = 1 \); and, because

\[ k^2 - k'^2 = k^2 - k'^2 = 1; \text{ we get } \frac{a^2}{k^2} + \frac{b^2}{k^2} = 1; \]

whence,

\[ 2k^2 = a^2 + b^2 - k^2 + \sqrt{(a^2 + b^2 - k^2)^2 + 4a^2 b^2}; \text{ and when } h \text{ is determined, then } k' = \sqrt{k^2 + b^2}. \]

In consequence of the equation \( \frac{a^2}{k^2} + \frac{b^2}{k^2} = 1 \), we may suppose \( a = h \sin. m \), and \( b = k' \cos. m \); let \( a' = k \sin. m \), and \( b' = k' \cos. m \); or \( a' = \frac{k}{k'} a \), and \( b' = \frac{k}{k'} b \); then the point determined by the co-ordinates \( a' \) and \( b' \) will be in the surface of the given spheroid, and consequently Attraction.

Let \( M \) denote the mass of the spheroid of which the axis is \( h \); and let \( e^2 = \frac{h^2 - k^2}{h^2} = \frac{k^2 - k'^2}{h^2} \); then (25) the attractions of this spheroid upon the point within its surface, determined by the co-ordinates \( a \) and \( b \), are these, viz.

That perpendicular to the equator, equal to

\[ a' \times \frac{3M'}{h^2 e^2} \left[ e' - \text{arc. tan. } e' \right]; \]

and that perpendicular to the axis, equal to

\[ b' \times \frac{3M'}{2h^2 e^2} \left\{ \text{arc. tan. } e' - \frac{e'}{1 + e'^2} \right\}. \]

But (26, Cor. 3) the attractions of the given spheroid, whose semiaxes are \( k \) and \( k' \), upon the point without its surface determined by the co-ordinates \( a \) and \( b \), will be found by multiplying the preceding expressions respectively by \( \frac{k^2}{k^2} \) and \( \frac{kk'}{k'^2} \). Let \( M \) be the mass of the given spheroid; then \( M = \frac{k^2 h}{M'} \); consequently \( \frac{k^2}{M'} = \frac{M}{M'} \cdot \frac{h}{k} = \frac{M}{M'} \cdot \frac{a}{a'} \); and \( \frac{kk'}{k'^2} = \frac{M}{M'} \cdot \frac{k}{k'} = \frac{M}{M'} \cdot \frac{b}{b'} \); wherefore the attractions of the given oblate spheroid upon a point without the surface determined by the co-ordinates \( a \) and \( b \), are as follows, viz.

The attraction perpendicular to the equator, equal to

\[ a \times \frac{3M}{h^2 e^2} \left[ e' - \text{arc. tan. } e' \right]; \]

and that perpendicular to the axis, equal to

\[ b \times \frac{3M}{2h^2 e^2} \left\{ \text{arc. tan. } e' - \frac{e'}{1 + e'^2} \right\}. \]

Cor. In the very same manner we may determine the attractions of an oblong spheroid of revolution upon a point without the surface.

Attractions of ellipsoids.

28. Let AMBN be one of the principal sections of an ellipsoid, C the centre, AB and MN the axes, D a point in the periphery of the section, and DO perpendicular to MN (Plate CII. fig. 18); the attraction of the ellipsoid upon a particle placed at the pole A is to the force with which a particle placed at D is attracted in the direction DO, as AC to DO.

Draw DFG perpendicular to AB, and through F describe an ellipsoid similar to the given ellipsoid, and similarly situated, and having the same centre. Conceive an indefinitely great number of planes, making indefinitely small angles with one another, to be drawn through DG, so as to divide the two ellipsoids into an indefinitely great number of thin solids or slices; then the sections of the ellipsoids made by every one of the planes will be similar and concentric ellipses, each of them having an axis perpendicular to DG (5). Wherefore the attractions of the elements of the ellipsoid FHKL, upon a particle at F, are respectively equal to the attractions of the elements of the ellipsoid AMBN, upon a particle at D in the direction DO (19). Wherefore the whole attraction of the ellipsoid FHKL, upon a particle at F, is equal to the attraction of the ellipsoid AMBN, upon a particle at D, in the direction DO. But the attractions of the ellipsoids AMBN and FHKL, upon particles at A and F, are to one another as AC to CF (8). Wherefore the attraction of the ellipsoid AMBN, upon a particle at the pole A, is to the force with which it attracts a particle at D in the direction DO, as AC to DO.

29. The attractions of ellipsoids upon particles placed in the surface, urging them in directions perpendicular to any attraction of the principal sections, are proportional to the distances of the particles from that section.

Let AMBN be one of the principal sections of an ellipsoid, C the centre, AB and MN the axes of the section, and P a point in the surface of the solid: the attraction of the ellipsoid upon a particle at the pole A (Plate CII. fig. 19) is to the force with which a particle at P is attracted in a direction parallel to AB, as the semiaxis AC is to the distance of P from the principal section perpendicular to AC.

Draw PDQ perpendicular to the section AMBN, and let it meet the surface again in Q; through D describe an ellipsoid similar to AMBN, similarly situated, and having the same centre; and through P draw a section SPRQ perpendicular to AB. As before, divide the solids into an indefinitely great number of thin slices, by planes drawn through PQ: the sections made by every one of those planes will be similar, and concentric ellipses having an axis of each perpendicular to PQ (5). Wherefore the attractions of the elements of the ellipsoid AMBN, upon a particle at P, in a direction perpendicular to the plane PRQS, are respectively equal to the attractions of the elements of the ellipsoid FHKL upon a particle at D, in a direction perpendicular to the same plane (19). Wherefore the attraction of the ellipsoid AMBN, upon a particle at P, in a direction parallel to the axis AB, is equal to the attraction of the ellipsoid FHKL upon a particle at D in the same direction. But the ellipsoids AMBN and FHKL being similar, their attractions upon particles at A and F are to one another as AC to CF (8); and the attraction of the ellipsoid FHKL, upon a particle at the pole F, is to its attraction upon a particle at D, in a direction parallel to AC, as FC to CN (28). Wherefore (ex aequo) the attraction of the ellipsoid AMBN, upon a particle at the pole A, is to the force with which it attracts a particle at P, in the direction AC, as AC to CN.

This proposition will enable us to find the attractions of an ellipsoid on all points on the surface, or within the solid, when the attractions at the poles are determined.

30. To investigate the differential expressions of the attractions at the poles of an ellipsoid.

Let APD be an ellipsoid; C the centre; AC, CE, and PC, the semiaxes; and PMB a section made by a plane through PC and any point M in the surface; draw PM (fig. 20) PM indefinitely near PM, and MM perpendicular to PM; also MR perpendicular to the plane ADB, MS perpendicular to PC, and RH perpendicular to AD. Conceive the plane PCB to revolve about PC, so as to describe an indefinitely small angle BCO; and let PM = \( f \); the angle KPM which PM makes with a perpendicular to the axis = \( \phi \); and the angle DCB = \( \varphi \); then, by proceeding as in (23), it will be found that the attraction of the small pyramid described by the triangle MPm, urging a particle at P to the centre of the ellipsoid, is \( d\varphi \cdot d\varphi \cos. \theta \sin. \theta \cdot f \).

Again, let MR = \( x \), HR = \( y \), CH = \( z \); also let PC = \( k \), AC = \( k \), CE = \( k' \); then \( x = k - f \sin. \theta \); \( y = f \cos. \theta \sin. \varphi \); \( z = f \cos. \theta \cos. \varphi \); and if we substitute these values in the equation of the ellipsoid (Def. 2), we shall get

\[ \frac{(k - f \sin. \theta)^2}{k^2} + \frac{f^2 \cos^2 \theta \sin^2 \varphi}{k^2} + \frac{f^2 \cos^2 \theta \cos^2 \varphi}{k^2} = 1; \]

whence

\[ f = \frac{k}{k^2 \sin^2 \theta + \cos^2 \theta \sin^2 \varphi + \cos^2 \theta \cos^2 \varphi}; \]

This is the value of \( f \) at the pole of \( k \); and, by a like procedure, its values at the poles of \( k \) and \( k' \) may be found, viz. Attraction.

\[ f = \frac{k}{k^2} \sin^2 \theta + \cos^2 \phi \sin^2 \phi + \cos^2 \theta \cos^2 \phi \]

\[ f' = \frac{k'}{k'^2} \sin^2 \theta + \cos^2 \phi \sin^2 \phi + \cos^2 \theta \cos^2 \phi \]

Suppose that \( k \) is the least of the semiaxes; and let

\[ k^2 = \frac{h^2}{m}, \quad k'^2 = \frac{h'^2}{n}; \]

then the values of \( f \) at the poles of \( h, k, k' \), will be respectively

\[ f = \frac{2k \sin \theta}{\sin^2 \theta + m \cos^2 \phi \sin^2 \phi + n \cos^2 \theta \cos^2 \phi} \]

\[ f' = \frac{2k' \sin \theta}{m \sin^2 \theta + \cos^2 \phi \sin^2 \phi + n \cos^2 \theta \cos^2 \phi} \]

Now let \( A, A', A'' \) denote the attractions of the spheroid upon particles placed at the poles of \( h, k, k' \); then, by substituting the values of \( f \) just found in the foregoing differential expression, we get

\[ A = k \times \int \int \frac{2d \phi d \cos \theta \sin^2 \phi}{\sin^2 \theta + m \cos^2 \phi \sin^2 \phi + n \cos^2 \theta \cos^2 \phi} \]

\[ A' = k' \times \int \int \frac{2d \phi d \cos \theta \sin^2 \phi}{m \sin^2 \theta + \cos^2 \phi \sin^2 \phi + n \cos^2 \theta \cos^2 \phi} \]

\[ A'' = k'' \times \int \int \frac{2d \phi d \cos \theta \sin^2 \phi}{n \sin^2 \theta + \cos^2 \phi \sin^2 \phi + m \cos^2 \theta \cos^2 \phi} \]

the limits of the integrals being from \( \theta = 0 \) and \( \phi = 0 \) to \( \theta = \frac{\pi}{2} \) and \( \phi = 2\pi \).

31. To reduce the expressions of the polar attractions to the most simple integrals.

Let us consider the general expression

\[ \int \int \frac{d \phi d \cos \theta \sin^2 \phi}{p \sin^2 \phi + q \cos^2 \phi} \]

which includes all the formulas found in (30). Let

\[ p = \alpha \sin^2 \theta + \beta \cos^2 \phi, \quad q = \gamma \sin^2 \theta + \gamma \cos^2 \phi; \]

then the above expression will become

\[ \int \int \frac{d \phi d \cos \theta \sin^2 \phi}{p \sin^2 \phi + q \cos^2 \phi} \]

Suppose \( \sqrt{\left(\frac{q}{p}\right)} \cdot \frac{\sin u}{\cos \phi} = \frac{\sin u}{\cos \phi} \); then the preceding expression will become, by substitution,

\[ \int \int \frac{du d \cos \theta \sin^2 \phi}{\sqrt{p q}}; \]

the limits of \( u \) being from 0 to \( 2\pi \); wherefore, by integrating with regard to \( u \), and restoring the values of \( p \) and \( q \), the integral becomes

\[ 2 \int \int \frac{d \phi d \cos \theta \sin^2 \phi}{\sqrt{(a \sin^2 \theta + b \cos^2 \phi)(c \sin^2 \theta + d \cos^2 \phi)}} \]

and, by putting \( x = \sin \theta \), the integral, which is to be taken from \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \) or from \( x = 0 \) to \( x = 1 \), will become

\[ 2 \int \int \frac{x^2 dx}{\sqrt{[a + (b - c)x^2][c + (d - a)x^2]}} \]

If now we take \( a, b, c, d \) so as to make the assumed expression coincide with the quantities \( A, A', A'' \), respectively, we shall get

\[ A = 4 \pi k \int \frac{x^2 dx}{\sqrt{[m + (1 - m)x^2][n + (1 - n)x^2]}} \]

\[ A' = 4 \pi k' \int \frac{x^2 dx}{\sqrt{[m + (1 - m)x^2][n + (1 - n)x^2]}} \]

These expressions have the inconvenience of containing different factors in the denominators; but they may be reduced to others having the same factors, by putting \( x = \frac{\tau}{\sqrt{m + (1 - m)\tau^2}} \) in the second, and \( x = \frac{\tau}{\sqrt{n + (1 - n)\tau^2}} \) in the third; we thus get

\[ A = 4 \pi k \int \frac{x^2 dx}{\sqrt{[m + (1 - m)x^2][n + (1 - n)x^2]}} \]

\[ A' = 4 \pi k' \int \frac{x^2 dx}{\sqrt{[m + (1 - m)x^2][n + (1 - n)x^2]}} \]

\[ A'' = 4 \pi k'' \int \frac{x^2 dx}{\sqrt{[m + (1 - m)x^2][n + (1 - n)x^2]}} \]

Now let

\[ \frac{1-m}{m} = \frac{k^2-k'^2}{k'^2} = \lambda^2, \quad \frac{1-n}{n} = \frac{k'^2-k''^2}{k''^2} = \lambda'^2; \]

Also let the mass of the ellipsoid \( M = \frac{4 \pi k k' k''}{3} \)

\[ = \frac{4 \pi k^2}{3 \sqrt{mn}}; \quad \text{then} \quad \frac{3M}{k^2} = \frac{4 \pi}{\sqrt{mn}}; \quad \text{wherefore, by substitution, we get} \]

\[ A = k \cdot \frac{3M}{k^2} \int \frac{x^2 dx}{\sqrt{[1 + \lambda^2 x^2][1 + \lambda'^2 x^2]}} \]

\[ A' = k' \cdot \frac{3M}{k'^2} \int \frac{x^2 dx}{\sqrt{[1 + \lambda^2 x^2][1 + \lambda'^2 x^2]}} \]

\[ A'' = k'' \cdot \frac{3M}{k''^2} \int \frac{x^2 dx}{\sqrt{[1 + \lambda^2 x^2][1 + \lambda'^2 x^2]}} \]

the integrations extending from \( x = 0 \) to \( x = 1 \).

These integrals cannot be expressed in finite terms. When \( \lambda \) and \( \lambda' \), or the eccentricities of the ellipsoid, are small, the values of the integrals may easily be found to a sufficient degree of exactness by series. They may likewise be all expressed by means of this fluent, viz.

\[ F = \int \frac{dx}{\sqrt{(1 + \lambda^2 x^2)(1 + \lambda'^2 x^2)}} \quad \text{(from } x = 0 \text{ to } x = 1) \]

and its partial fluxions. Thus we have, in general,

\[ \int \frac{x^2 dx}{\sqrt{(1 + \lambda^2 x^2)(1 + \lambda'^2 x^2)}} = \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right) \]

Therefore, making \( x = 1 \), we get

\[ A = k \cdot \frac{3M}{k^2} \left( \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right) \right) \]

\[ A' = k' \cdot \frac{3M}{k'^2} \left( \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right) \right) \]

\[ A'' = k'' \cdot \frac{3M}{k''^2} \left( \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right) \right) \]

32. To find the forces with which a homogeneous ellipsoid attracts a particle placed in the surface, or within the solid, in directions perpendicular to the principal sections.

Let \( h, k, k' \) denote the semiaxes of an ellipsoid, and \( a, b, c \) (respectively parallel to \( h, k, k' \)) the perpendicular distances of a particle placed in the surface, or within Attraction, the solid, from the principal sections; then, from what is proved in (29), the attractions we are seeking will be found by multiplying the polar attractions by \( \frac{a}{k} \cdot \frac{b}{k'} \cdot \frac{c}{k''} \).

Wherefore the forces that urge the particle in the directions of \( a, b, \) and \( c \), are respectively \( a \times \frac{3M}{k^3} \):

\[ \left\{ \frac{1}{\sqrt{(1 + \lambda^2)(1 + \lambda'^2)}} + \frac{1}{\lambda} \left( \frac{dF}{dx} \right) + \frac{1}{\lambda'} \left( \frac{dF}{dy} \right) \right\}; \]

\( b \times \frac{3M}{k^3} \cdot \frac{1}{\lambda} \cdot \left( \frac{dF}{dx} \right) \);

and \( c \times \frac{3M}{k^3} \cdot \frac{1}{\lambda'} \cdot \left( \frac{dF}{dy} \right) \).

Which formulas serve both for points in the surface and within the solid, for the reason already explained in (25).

33. To find the attractions of an ellipsoid upon a particle placed without the surface.

Let \( k, k', k'' \) be the semiaxes of the ellipsoid, and \( a, b, c \) (respectively parallel to \( k, k', k'' \)) the co-ordinates of a particle without the surface. Let \( h, h', h'' \), so related to \( k, k', k'' \), that \( h^2 - k^2 = k'^2 - k^2 \) and \( h'^2 - k'^2 = k''^2 - k'^2 \), denote the semiaxes of another ellipsoid, which contains the attracted point in its surface, and has its principal sections in the same planes as the given ellipsoid; then, because the attracted point is in the surface, we have (Def. 2)

\[ \frac{a^2}{k^2} + \frac{b^2}{k'^2} + \frac{c^2}{k''^2} = 1; \]

and, because \( h^2 - k^2 = k'^2 - k^2 = e^2 \), and \( h'^2 - k'^2 = k''^2 - k'^2 = e'^2 \),

This equation now contains only one unknown quantity; and it is plain that one value of \( h \), and only one, can be found from it. For, when \( h = 0 \), the function on the left hand side is infinitely great; and while \( h \) increases from 0 ad infinitum, the same function decreases continually from being infinitely great to be infinitely little. When \( h \) is found, then \( h' = \sqrt{k'^2 + e'^2} \), and \( h'' = \sqrt{k''^2 + e'^2} \). Because \( a, b, c \), are the co-ordinates of a point in the surface of the ellipsoid, we may suppose \( a = h \sin m, b = k' \cos m \sin n, c = k'' \cos m \cos n \); let \( a' = k \sin m, b' = k' \cos m \sin n, c' = k'' \cos m \cos n \); or \( a' = \frac{k}{k'} \times a, b' = \frac{k}{k'} \times b, c' = \frac{k}{k'} \times c \); then \( a', b', c' \), will be the co-ordinates of a point in the surface of the given ellipsoid, and consequently it will be within the other solid. Let \( M' \) denote the mass of the ellipsoid of which \( h, h', h'' \), are the semiaxes; also let \( \lambda^2 = \frac{h^2 - k^2}{k^2} = \frac{h'^2 - k'^2}{k'^2} \); and \( \lambda'^2 = \frac{h''^2 - k'^2}{k'^2} \); then, \( F \) denoting the same fluent as before, the attractions of this ellipsoid upon the point within it, determined by the co-ordinates \( a', b', c' \), in the directions of those co-ordinates, are (32) respectively equal to

\[ a' \times \frac{3M'}{k^3} \left\{ \frac{1}{\sqrt{(1 + \lambda^2)(1 + \lambda'^2)}} + \frac{1}{\lambda} \left( \frac{dF}{dx} \right) + \frac{1}{\lambda'} \left( \frac{dF}{dy} \right) \right\}; \]

\( b' \times \frac{3M'}{k'^3} \cdot \frac{1}{\lambda} \cdot \left( \frac{dF}{dx} \right) \);

and \( c' \times \frac{3M'}{k''^3} \cdot \frac{1}{\lambda'} \cdot \left( \frac{dF}{dy} \right) \).

Now, the attractions of the given ellipsoid upon the point without the surface, determined by the co-ordinates \( a, b, c \), will be found (26) by multiplying the preceding expressions respectively by \( \frac{k}{k'}, \frac{k}{k''}, \frac{k}{k''} \). Let \( M \) be the mass of the given ellipsoid; then \( \frac{M}{M'} = \frac{k}{k'}, \frac{k}{k''}, \frac{k}{k''} \); consequently \( \frac{k}{k'} \cdot \frac{k}{k''} \cdot \frac{k}{k''} = \frac{M}{M'} \cdot \frac{a}{a'}, \frac{k}{k''} \cdot \frac{M}{M'} \cdot \frac{b}{b'} \); and \( \frac{k}{k'} \cdot \frac{k}{k''} \cdot \frac{k}{k''} = \frac{M}{M'} \cdot \frac{c}{c'} \); wherefore the attractions of the given ellipsoid upon the point without the surface, determined by \( a, b, c \), in the directions of those co-ordinates, are respectively equal to \( a \times \frac{3M}{k^3} \times \)

\[ \left\{ \frac{1}{\sqrt{(1 + \lambda^2)(1 + \lambda'^2)}} + \frac{1}{\lambda} \left( \frac{dF}{dx} \right) + \frac{1}{\lambda'} \left( \frac{dF}{dy} \right) \right\}; \]

\( b \times \frac{3M}{k^3} \cdot \frac{1}{\lambda} \cdot \left( \frac{dF}{dx} \right) \), and \( c \times \frac{3M}{k^3} \cdot \frac{1}{\lambda'} \cdot \left( \frac{dF}{dy} \right) \).

The preceding propositions contain a complete theory of homogeneous elliptical spheroids. They enable us to compute the attractive force with which a solid of this kind urges a particle placed anywhere in the surface, within the solid, or without it. It remains, indeed, to find the exact value of the function \( F \) in its general form, to which we can do no more than approximate by series; but this is an analytical difficulty which it is impossible to overcome, because the nature of this function is such that it cannot be expressed in finite terms by the received notation of analysis.

In the preceding investigations we have followed the method of Maclaurin for points situated in the surface of a spheroid or within the solid. This method has always been justly admired; but neither its inventor, nor, as far as we know, any other geometer, has applied it, excepting to spheroids of revolution; and it is here, for the first time, extended to ellipsoids. In regard to points without the surface, we have employed the method first given by Mr Ivory, in the Philosophical Transactions for 1809. The combination of these two methods has enabled us to derive the attractions of an ellipsoid on a point placed anywhere from the attractions at the poles. Thus, this extremely complicated problem has, by geometrical reasoning of no great difficulty, been reduced to the investigation of the polar attractions, which are the only cases that require a direct computation.

34. Of the attractions of spheroids composed of elliptical shells that vary in their densities and figures according to any law.

When a spheroid is composed of concentric elliptical shells of variable density and figure, we may consider every shell as the difference of two homogeneous spheroids of the same density with the shells, and having their surfaces coinciding with the surfaces of the shell. The attractions of the spheroids being computed by the preceding methods, their difference will be equal to the attractions of the shell, and the integral obtained by summing the attractions of all the shells will give the attractions of the heterogeneous spheroid. This case, therefore, gives rise to no new difficulties, except such as are purely mathematical, and depend upon the law according to which the densities and figures of the shells are supposed to vary.