117. Results from Volume of Ordnance Survey 118. Process of finding the Ellipse which best suits the ob- ib. ។ ERRATA. Page 1, 1. 8, for nearly circular, read nearly spherical 12, 1. 2, for ¥ (r) = }ƒ¢(r) dr: read ¥(r)=}ƒ¢(r)dr: 91, 1. 21, dele from The general to theory 1. ATTRACTIONS AND LAPLACE'S FUNCTIONS. 1. THE Law of Universal Gravitation teaches us, that every particle of matter in the universe attracts every other particle of matter with a force varying directly as the mass of the attracting particle and inversely as the square of the distance between the attracted and the attracting particles. Taking this law as our basis of calculation, we shall investigate the amount of attraction exerted by spherical, spheroidal, and irregular nearly-circular masses upon a particle, and apply our results in the second part of this Treatise to discover the Figure of the Earth. We shall also show how the attraction of irregular masses lying at the surface of the Earth may be found in order to estimate the effect of the irregularities of mountain-land and the ocean in modifying this figure. CHAPTER I. ON THE ATTRACTION OF SPHERICAL AND SPHEROIDAL BODIES. PROP. To find the resultant attraction of an assemblage of particles constituting a homogeneous spherical shell of very small thickness upon a particle outside the shell: the law of attraction of the particles being that of the inverse square. The attraction of the whole shell evidently acts in CO. Let OP revolve about O through a small angle de in the plane MOP; then rde is the space described by P. Again, let OPM revolve about OC through a small angle do, then r sin do is the space described by P. And the thickness of the shell is dr. Hence the volume of the elementary portion of the shell thus formed at P equals rde .r sin Odp.dr ultimately, since its sides are ultimately at right angles to each other. Then, if the unit of attraction be so chosen, that it equals the attraction of the unit of mass at the unit of distance, the attraction of the elementary mass at P on C in the direction CP = pr2 sin 0 drdedo , p the density of the shell; .. attraction of P on C in CO = p2 sine drdedo c-r cos y2 y We shall eliminate @ from this equation by means of To obtain the attraction of all the particles of the shell we integrate this with respect too and y, the limits of being 0 and 2π, those of y being c .. attraction of shell on C: πρ = ctr = c2 r and c+r; -"prdr [(1 + 2 =) dy = "prdr (2r +2r) y2 4πpr2dr mass of shell c2 = = c2 This result shows that the shell attracts the particle at C in the same manner as if the mass of the shell were condensed into its centre. 3. It follows also that a sphere, which is either homogeneous or consists of concentric spherical shells of uniform density, will attract the particle C in the same manner as if the whole mass were collected at its centre. PROP. To find the attraction of a homogeneous spherical shell of small thickness on a particle situated within it. 4. We must proceed as in the last Proposition; but the limits of y are in this case r -c and r+c; hence, therefore the particle within the shell is equally attracted in every direction. 5. This result may easily be arrived at geometrically in the following manner. Through the attracted point suppose an elementary double cone to be drawn, cutting the shell in two places. The inclinations of the elementary portions of the shell, thus cut out, to the axis of the cone will be the same, the thickness the same, but the other two dimensions of the elements will each vary as the distance from the attracted point; and therefore the masses of the two opposite elements of the shell will vary directly as the square of the distance from that point, and consequently their attractions will be exactly equal, and being in opposite directions will not affect the point. The whole shell may be thus divided into pairs of equal attracting elements and in opposite directions, and therefore the whole shell has no effect in drawing the point in any one direction more than in another. 6. The results of these two Propositions are so simple and beautiful, that it is interesting to enquire whether these properties belong exclusively or not to the law of the inverse square of the distance. To determine this is the object of the four following Propositions. PROP. To find the attraction of a homogeneous spherical shell on a particle without it; the law of attraction being represented by (y), y being the distance. 7. The calculation is exactly analogous to that given above: we have only to alter the law of attraction. Then attraction on Cin CO = = = ctr "prdr [*** (y2 + c2 — r2) $ (y) dy (integrated by parts) c2 προαν c2 πprdr c2 [(y2 + c2 — r2) S$ (y) dy — 2ƒ{yfþ(y) dy} dy] {(y2 + c2 — r2) 4, (y) — 2¥ (y) + const.} suppose, 1 =2πprdr {0+* 4, (c+r) — — ↓ (c+r). =2πрrdr C - Þ2 (c+r) — — a ¥ (c+r) − ∞—” $, (0–r) + && (c–r)} this latter form being introduced merely as an analytical artifice to simplify the expression. PROP. To find the attraction of the shell on an internal particle, with the same law. 8. The calculation is the same as in the last Article, except that the limits of y are r―c and r+c: .. attraction = 2æprdr {~ +° 4, (r+c) − † † (r + c) |