AT THE SURFACE OF THE EARTH. 695 



— 2/3, — j3, 0, /3, 2/3 ..., mark these points on a map of the world, and join by a curve those 

 which belong to the same value of An. We shall thus have a series of contour lines representing 

 the elevation or depression of the actual sea-level above or below the surface of the oblate spheroid, 

 which has been employed as most nearly representing it. If we suppose these lines traced on a 

 globe, the reciprocal of the perpendicular distance between two consecutive contour lines will 

 represent in magnitude, and the perpendicular itself in direction, the deviation of the vertical from 

 the normal to the surface of the spheroid, or rather that part of the deviation which takes place on 

 an extended scale : for sensible deviations may be produced by attractions which are merely local, 

 and which would not produce a sensible elevation or depression of the sea-level ; although of course, 

 as to the merely mathematical question, if the contour lines could be drawn sufficiently close and 

 exact, even local deviations of the vertical would be represented. 



Similarly, by joining points at which the quantity denoted in Art. 19 by V has a constant 

 value, contour lines would be formed representing the elevation of the actual sea-level above what 

 would be a surface of equilibrium if the earth's irregular coating were removed. By treating F, 

 in the same way, contour lines would be formed corresponding to the elevation of the actual sea- 

 level above what would be the sea-level if the solid portions of the earth's crust which are 

 elevated were to become fluid and to run down, so as to form a level bottom for the sea, which 

 would in that case cover the whole earth. 



These points of the theory are noticed more for the sake of the ideas than on account of any 

 application which is likely to be made of them ; for the calculations indicated, though possible with 

 a sufficient collection of data, would be very laborious, at least if we wished to get the results 

 with any detail. 



ii5. The squares of the ellipticity, and of quantities of the same order, have been neglected 

 in the investigation. Mr. Airy, in the Treatise already quoted, has examined the consequence, on 

 the hypothesis of fluidity, of retaining the square of the ellipticity, in the two extreme cases of a 

 uniform density, and of a density infinitely great at the centre and evanescent elsewhere, and has 

 found the correction to the form of the surface and the variation of gravity to be insensible, or 

 all but insensible. As the connexion between the form of the surface and the variation of gravity 

 follows independently of the hypothesis of fluidity, we may infer that the terms depending on the 

 square of the ellipticity which would appear in the equations which express that connexion would 

 be insensible. It may be worth while, however, just to indicate the mode of proceeding when the 

 square of the ellipticity is retained. 



By the result of the first approximation, equation (l) is satisfied at the surface of the earth, 

 as far as regards quantities of the first order, but not necessarily further, so tliat the value of 

 r -I- ^7 at the surface is not strictly constant, but only of tlie form c + //, where H is a small 

 variable quantity of the second order. It is to be observed that V satisfies equation (.'i) exactly, 

 not approximately only. Hence we have merely to add to F a potential V which satisfies equation 

 (."ii outside the earth, vanishes at an infinite distance, and is equal to // at the surface. Now if 

 we suppose V to have the value // at the surface of a spliere whoso radius is a, instead of the 

 actual surface of the earth, we shall only conunit an error whicli is a small quantity of the first 

 order compared with J/, and // is itself of the second order, and therefore the error will be otdy 

 of the third order. But by this modification of one of the conditions which V is to satisfy, we 

 are enabled to find V just as V was found, and we shall thus have a solution which is correct to 

 the second order of ap])roxiniation. A repetition of the same process would give a solution 

 which would be correct to the third order, and so on. It ne(<l hartlly be remarked that in going 

 beyond the first order of approximation, we must distinguish in the small terms between the 

 direction of the vertical, and that of the radius vector. 



G. G. STOKKS. 

 4u S 



