AT THE SURFACE OF THE EARTH. 681 



conclusion cannot be drawn without fm-thtr consideration except on the supposition that the earth 

 is solid to the centre. If we assume this coincidence, the term £"'(i — cos° 0) will unite with the 

 term u" due to the centi-ifugal force. Thus the most general value of u is that which belongs to an 

 ellipsoid having one of its principal axes coincident with the axis of rotation, added to a quantity 

 which, if expanded in a series of Laplace's coefficients, would furnish no terms of the order 0, I, or 2. 

 It appears from this and the preceding article that the coincidence of the centres of gravity of the 

 mass and volume, and that of the axis of rotation and one of the principal axes of the ellipsoid whose 

 equation is r= a(l + M2), which was established by Laplace on the supposition that the earth consists of 

 nearly spherical strata of equal density, holds good whatever be the distribution of matter in the interior. 



1.3. Hitherto the surface of the earth has been regarded as a surface of equilibrium. This we 

 know is not strictly true, on account of the elevation of the land above the level of the sea. The 

 question now arises, By what imaginary alteration shall we reduce the surface to one of equilibrium;' 



Now with respect to the greater portion of the earth's surface, which is covered with water, we 

 have a surface of equilibrium ready formed. The expression level nf the sea has a perfectly definite 

 meaning as applied to a place in the middle of a continent, if it be defined to mean the level at 

 which the sea-water would stand if introduced by a canal. The surface of the sea, supposed to be 

 prolonged in the manner just considered, forms indeed a surface of equilibrium, but the preceding 

 investigation does not apply directly to this surface, inasmuch as a portion of the attracting matter 

 lies outside it. Conceive however the land which lies above the level of the sea to be depressed till 

 it gets below it, or, which is the same, conceive the land cut off at the level of the sea produced, 

 and suppose the density of the earth or rock which lies immediately below the sea-level to he in- 

 creased, till the increase of mass immediately below each superficial element is equal to the mass 

 which has been removed from above it. The whole of the attracting matter will thus be brouglit 

 inside the original sea-level; and it is easy to see that the attraction at a point of space external to 

 the earth, even though it be close to the surface, will not be sensibly affected. Neither will tlie 

 sea-level be sensibly changed, even in the middle of a continent. For, suppose the sea-water intro- 

 duced by a pipe, and conceive the land lying above the sea-level condensed into an infinitely thin 

 layer coinciding with the sea-level. The attraction of an infinite plane on an external particle does 

 not depend on the distance of the particle from the plane ; and if a line be drawn througli the 

 particle inclined at an angle a to the perpendicular let fall on the plane, and be then made to revolve 

 around the perpendicular, the resultant attraction of the portion of the plane contained witliin the 

 cone thus formed will be to that of the whole plane as versin a to 1. Hence the attraction of a 

 piece of table-land on a particle close to it will be sensibly the same as that of a solid of equal 

 thickness and density comprised between two parallel infinite planes, and that, even though the 

 lateral extent of the table-land be inconsiderable, only equal, suppose, to a small multiple of the 

 lengtii of a perpendicular let fall from the attracted particle on the further bounding phme. Ilenie 

 the attraction of the land on the water in the tube will not be sensibly altered by the condensation we 

 have supposed, and therefore we are fully justified in regarding the level of the .sea as unchanged. 



The surface of equilibrium which by the imaginary displacement of matter just considered has 

 also become the bounding surface, is that surface which at the same time coincides with the surface 

 of the actual sea, where the earth is covered by water, and belongs to the system of surfaces i>f 

 equilibrium which lie wholly outside the earth. 'J"o reduce observed gravity to what would have 

 been observed just above this imaginary surface, we n)Ust evidently increase it in the inverse ratio 

 of the square of the distance from the centre of the earth, without taking account of the attraction 

 (,f the table-land which lies between the level of the station and the level of the sea. The question 

 now arises. How shall we best determine the numerical value of the eartii's cllipticity, and how 

 best compare the form which results from observation with the spheroid whicli results from theory 

 on the hypi)the>is of original fluidity .' 



