684 Mr. stokes, ON THE VARIATION OF GRAVITY 



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 4 



111 considering the value of q we may attend only to the land, provided we transfer the defect of 

 density of the sea with an opposite sign to the land, because if g' were constant, q would vanish. 

 This of course proceeds on the supposition that the depth of the sea is constant. Since e = e, — q, 

 if q were positive, the ellipticity determined by the pendulum would appear too great in con- 

 sequence of the omission of the force g. I have made a sort of rough integration by means of 

 a map of the world, by counting the quadrilaterals of land bounded each by two meridians 

 distant 10", and by two parallels of latitude distant 10°, estimating the fraction of a broken 

 quadrilateral wliich was partly occupied by sea. The number of quadrilaterals of land between 

 two consecutive parallels, as for example 50° and 60", was multiplied by 12 (1^ - sin'/) cos /, or 

 .S cos 3 / + cos /, where for / was taken the mean latitude, (S5° in the example,) the sum of the 

 results was taken for the whole surface, and multiplied by the proper coefficient. The north pole 

 was supposed to be surrounded by water, and the south pole by land, as far as latitude 80''. It 

 appeared that the land lying beyond the parallels for which sin'7 = ^, that is, beyond the 

 parallels 35° N. and 35° S. nearly, was almost exactly neutralized by that which lay within those 

 parallels. On the whole, q appeared to have a very small positive value, which on the same 

 suppositions as before respecting the height of the land and the depth of the sea, was .0000012. 

 It appears, therefore, that the omission of the force g' will produce no sensible increase in the 

 value of £, unless the land be on the whole higher, or the sea shallower, in high latitudes than in 

 low. If the land had been collected in a great circular continent around one pole, the value of 

 7 would have been .000268 ; if it had been collected in a belt about the equator, we should have 

 had q = — .000362. The difference between these values of q is about one fifth of the whole 

 ellipticity. 



19. The attraction g' is not the only irregularity in the magnitude of the force of gravity 

 which arises from the irregularity in the distribution of land and sea, and in the height of the 

 land and depth of the sea, although it is the only irregulai'ity, arising from that cause, which is 

 liable to vary suddenly from one point at the surface to another not far off. The irregular coating 

 of the earth will produce an irregular attraction besides that produced by the part of this coating 

 which lies under and in the immediate neighbourhood of the station considered, and it will 

 moreover cause an irregular elevation or depression in the level of the sea, and thereby cause a 

 diminution or increase in the value of g^. 



Consider the attraction arising from the land which lies above the level of the sea, and from 

 the defect of attracting matter in the sea. Call this excess or defect of matter the coating of the 

 earth : conceive the coating condensed into a surface coinciding with the level of the sea, and 

 let Ac be the mass contained in a small element A of this surface. Then S = ah in the case of the 

 land, and ^ = — {a — I) h in the case of the sea, h being in that case the depth of the sea. Let 

 V^ be the potential of the coating, V', V' the values of V^ outside and inside the surface respec- 

 tively. Conceive S expanded in a series of Laplace's coefficients ^„ + ^i + ..., then it is easily 

 proved that 



V' = .,^a^(l§, + ^J, + fj,+ ..X V" = t^a'(-^, + ^J,+ ...], (28) 



\r 3r' 5r^ j \a 3a' J 



r being the distance of the point considered from the centre. These equations give 



dV ^ i + \ fa\**\ dF" ^ i try-' 



-— =-47r2-T - 4-^ = 4x2-^ - Si (29) 



dr 2t + 1 \rl ' dr 2j + 1 \a! 



Consider two points, one external, and the other internal, situated along the same radius vector 



