evection, the change of inclination of the lunar orbit, and the precession of the equinoxes; the proposition is also applied to the theory of the tides, and to the determination of the interior constitution of the earth as deduced from the motion of its nodes.

The general problem of the motion of three bodies under their mutual attraction still remains unsolved, and that Newton should have been able, with the limited analysis at his command, to work it out so far in the case of the moon, is worthy of special notice.

Prop. 67. Under the same hypotheses as in prop. 66, and if O be the centre of gravity of P and T, then the areas described by S round O will be more nearly proportional to the times of description than the areas described by S round T, and the orbit of P will approximate more nearly to an ellipse with O as focus than to an ellipse with T as focus.

Prop. 68. Under the same hypotheses, the areas described by S round O will be more nearly proportional to the times of description and the orbit of S about O will approximate more nearly to an ellipse with O as focus than would be the case if T were not attracted by S and P but remained at rest, or if T were attracted (or moved) very much more or very much less.

Prop. 69. The absolute force of any one of a system of attracting bodies is, under the usual hypotheses, proportional to the mass of the body.

Scholium. The above propositions naturally lead to the consideration as to how the attraction of a body depends on its form; this is to be determined by summing the attractions of all its component particles, and to effect this it is not necessary to propound a theory as to how attraction is produced, whether "ab actione corporum vel se mutuo petentium, vel per Spiritus emissos se invicem agitantium; sive is "ab actione Ætheris aut Eris mediive cujuscunque seu corporei seu incorporei oriatur corpora innatantia in se invicem utcunque impellentis. In Mathesi investigandae sunt virium quantitates et "rationes illae, quae ex conditionibus quibuscunque positis consequentur: deinde ubi in Physicam descenditur, conferendae sunt hae "rationes cum Phaenomenis.”

66

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SECTION XII.-On the attractions of spherical bodies.

,

Prop. 70. If to every point of a spherical surface there tend a force varying inversely as the square of the distance, and if all these forces be of equal absolute magnitude, then the resultant on a particle inside the surface is nothing, that is, a particle placed anywhere inside a homogeneous spherical shell is in equilibrium.

Prop. 71. Under the same hypotheses a particle outside the shell is attracted to the centre of the shell with a force inversely proportional to the square of the distance from that centre.

Prop. 72. If to every point of a solid sphere centre S, radius r, and of given density, there tend a force varying inversely as the square of the distance, and if all these forces be of equal absolute magnitude, and if the distance of a particle P from S be a given multiple of r, then the attraction of the sphere on P is proportional to r.

Prop. 73. If to every point of a solid sphere centre S there tend a force varying inversely as the square of the distance, and if all these forces be of equal absolute magnitude, then the attraction of the sphere, on a particle P inside it, is proportional to SP.

Scholium. On the physical meaning to be attached to the points of which lines, surfaces, and solids are said to be composed.

Prop. 74. Under the same hypotheses the attraction of the sphere, on a particle P outside it, is inversely proportional to SP2.

Prop. 75. Under the same hypotheses the attraction of the sphere, centre S, on another similar sphere, centre P, is inversely proportional to SP2.

Prop. 76. If to every point of a sphere, centre S, there tend a force varying inversely as the square of the distance, and if the absolute magnitude of the force at any point A and the density at A be functions only of SA, then the attraction of the sphere on another such sphere, centre P, is inversely proportional to SP2.

To this proposition were appended nine corollaries on the motion of such spheres about one another in conics.

Prop. 77. If to all the points of [homogeneous] spheres there tend forces directly proportional to the distances, then two spheres, centres S and P, attract each other with a force proportional to SP.

Prop. 78. The result of prop. 77 is true if the spheres be dissimilar and heterogeneous, provided the density at any point of a sphere is a function only of its distance from the centre of that sphere.

Scholium. The above propositions cover the two most important cases of attractions; other cases can be deduced from the general method explained in the following propositions. [It is arguable that the Newtonian method here indicated is not less powerful than direct analytical methods; see Brougham and Routh on the Principia, pp. 140-145.]

Lemma 29. If two points S and P be taken; and if about S there be described any circle [whose radius is less than SP], and about P there be also described two circles very close to one another, cutting the circle about S in E and e (situated on the same side of SP), and cut

ting the line PS in F and ƒ; and there be let fall to PS the perpendiculars ED, ed; then, in the limit when e coincides with E, and ƒ with F, the ultimate ratio of Dd to Ff is equal to the ratio of PE to PS.

Prop. 79. Determination of the attraction of the segment of an infinitely thin homogeneous spherical shell on a particle at its centre under any law of force.

Prop. 80. Determination of the attraction of a solid homogeneous sphere on an external particle under any law of force. The result is given as a multiple of a certain area.

Prop. 81. Under the same hypotheses, to find this area.

Newton then applies the results of props. 80, 81 to the cases where the force is μr and (i) n= −1, (ii) n= — = −3, (iii) n=

-4.

Prop. 82. Determination of the attraction of a solid homogeneous sphere on an internal particle. This is found as a multiple of the attraction on an external inverse particle.

Prop. 83. To find the force with which a segment of a sphere attracts a particle placed at the centre of the sphere; the force varying inversely as the nth power of the distance.

Prop. 84. To find the force with which a segment of a sphere attracts a particle placed at a point on the axis of the segment; the force varying inversely as the nth power of the distance.

Scholium. The attractions of non-spherical bodies next require attention.

SECTION XIII.-On the attractions of non-spherical bodies.

Prop. 85. If the attraction of a body on a contiguous body be much greater than on the same body when it is separated from the attracting body by a small interval; then the attractive forces of the particles of the attracting body decrease in a higher ratio than the inverse square of the distance.

Prop. 86. If the attractive forces of the particles of a body vary as the inverse cube of the distance (or in a higher ratio), then the attraction of the body on a contiguous body at the point of contact is much greater than it would be if the attracting and attracted bodies be separated from each other, though by ever so small an interval.

Prop. 87. If two similar bodies of the same material attract separately two particles whose masses are proportional to those bodies, and which are similarly situated to them, then the attractions of the particles on the bodies will be proportional to the attractions of the particles towards particles of the bodies whose masses are proportional to the bodies and which are similarly situated in them.

Prop. 88. If the particles of any body, whose centre of gravity is G,

attract with forces directly proportional to the distance, then the attraction of the body on any particle Z will tend to G, and will be the same as that of a sphere of equal and similar matter whose centre is G.

Prop. 89. The result of prop. 88 is true also for a system of bodies. Prop. 90. To find the attraction of a uniform circular lamina on a particle placed on its axis, under any law of attraction. This is applied in cor. 1 to the case when the law is μ/2, in cor. 2 to the case when the law is μ/rn, and in cor. 3 to the case of an infinite plate when the law is μ/ and n>1.

Prop. 91. To find the attraction of a solid of revolution on a particle placed on its axis, under any law of attraction. This is applied in cor. 1 to a cylinder when the law of force is μ/r2, in cor. 2 to a spheroid on an external axial particle, and in cor. 3 to a spheroid on an internal axial particle.

Prop. 92. To find by experiment the law of attraction of the particles of a given body.

Prop. 93. If an infinite homogeneous solid terminated on one side by a plane be composed of particles which attract with a force which varies inversely as the nth power of the distance, where n is greater than 2, then the attraction of the solid on a particle is inversely proportional to yn-3 where y is the distance of the particle from the plane.

Scholium. On the determination of the orbit described by a body attracted perpendicularly towards a given plane according to a given law; and conversely on the determination of the law of force under which a body will describe a given orbit.

Newton adds that if the equation of the [plane] orbit (referred to a line on the plane as axis of a and a line inclined to Ox at a fixed angle as axis of y) be given in such a form that the ordinate can be expanded in a convergent series, then the general method for finding the law of force parallel to the ordinates under which a body will describe the curve may be replaced by a simple rule which will be sufficiently illustrated by the case of the curve cy am/n. The ordinate at a point whose abscissa iso, where o is very small, is determined by the equation

ཡ

and the force required will be proportional to the coefficient of the term on the right-hand side involving o2. For example, in the parabola cy x2, we have m = 2, n = 1, and the force is constant; in the hyperbola xy d2 we have m = n = 1, and the force is proportional to x-3 that is to y3 [in fact, generally, if the curve be y = f(x), the force

- 1,

varies as y"]. But, says he, leaving propositions of this kind, I shall proceed to some others concerning a kind of motion which I have not yet discussed.

SECTION XIV.-On the motion of minute corpuscles when acted on by centripetal forces tending to the several parts of any large body.

Prop. 94. If two similar mediums be separated by a space contained between parallel planes, and a body in its passage through that space be attracted towards either of these bounding planes with a force depending only on the distance from the plane, and be not acted on by any other force; then the sine of the angle (Ø) of incidence upon the first plane will be to the sine of the angle (') of emergence from the second plane in a given ratio.

Prop. 95. Under the same hypotheses the velocity of the corpuscle before incidence is to the velocity after emergence as sino' to sino.

Prop. 96. Under the same hypotheses, and assuming that the velocity before incidence is greater than afterwards, then if the angle of incidence be increased continually, the corpuscle will be at last reflected, and the angle of reflexion will be equal to the angle of incidence.

Scholium. On the application of the above propositions to the theory of light, on the finite velocity of light, and on diffraction phenomena. Since there is an analogy between the propagation of the rays of light and the motion of bodies, Newton adds two propositions which are applicable to optics, and in establishing which it is unnecessary to consider the nature of the rays of light or whether the corpuscular theory is true.

Prop. 97. A system of rays diverge from a given point; to find an (aplanatic) surface which will refract them to a given point.

Prop. 98. A system of rays diverge from a given point, and are refracted at a surface of revolution about an axis through the point; to find an (aplanatic) surface which will refract them to a given point on the axis, i.e. to construct an aplanatic lens.

Scholium. The above methods are applicable when the rays are refracted at more surfaces. In constructing optical instruments it is preferable to use only lenses whose surfaces are spherical, not only because they can be made more readily and accurately, but also because rays incident obliquely would be refracted to a point more accurately than by spheroidal lenses. It is chromatism, however, that is the real obstacle to perfecting practical optics, and unless the errors thence arising can be corrected, "labor omnis in caeteris corrigendis imperite "collocabitur."