CONTENTS. 114. Mr Hopkins' investigation analysed 115. Remarks on calculations of Professors Hennessy, Haughton, PAGE 109 112 113 119 122. Length of an arc of latitude and one of longitude in terms of amplitude, semi-axes, and middle latitude 133. Degree of uncertainty thus introduced into the problem 134. Mean figure found by comparing the Anglo-Gallic, Russian, 137. Speculations regarding the constitution of the earth's crust. 152. Mean and geodetic arcs of longitude sensibly the same ATTRACTIONS AND LAPLACE'S 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-spherical 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 estimated, in order afterwards to ascertain whether the irregularities of mountain-land and the ocean can have any effect on the calculation of 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 edo 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 drd0dp , p the density of the shell; .. attraction of P on C in COP sin drded = y2 -r cos 0 Y We shall eliminate O from this equation by means of being To obtain the attraction of all the particles of the shell we integrate this with respect to p and y, the limits of 0 and 2π, those of y being c-r and c+r; SPHERICAL SHELL ON INTERNAL POINT. 3 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 |