692 Mr. stokes, ON THE VARIATION OF GRAVITY 



due to a very small area surrounding the station is very small. If S vary abruptly, in consequence 

 suppose of the occurrence of a cliff, we may employ the expressions (40), (41), provided the distance 

 of the cliff from the station be as much as three or four times its height. 



These expressions shew that the vertical is liable to very irregular deviations depending on 

 attractions which are quite local. For it is only in consequence of the opposition of attractions in 

 opposite quarters that the value of the integral is not considerable, and it is of course larger in 



proportion as that opposition is less complete. Since sin — is but small even at the distance of two 



or three hundred miles, a distant coast, or on the other hand a distant tract of high land of con- 

 siderable extent, may produce a sensible effect; although of course in measuring an arc of the 

 meridian those attractions may be neglected which arise from masses which are so distant as to affect 

 both extremities of the arc in nearly the same way. 



If we compare (40) or (41) with the expression for g' or g" , we shall see that the direction of 

 the vertical is liable to far more irregular fluctuations on account of the inequalities in the earth's 

 coatino- than the force of gravity, except that part of the force which has been denoted by g\ and 

 which is easily allowed for. It has been supposed by some that the force of gravity alters irregularly 

 along the earth's surface, and so it does, if we compare only distant stations. But it has been 

 already remarked with what apparent regularity gravity when corrected for the inequality g appears 

 to alter, in the direction in which we should expect, in passing from one station to another in a 

 chain of neighbouring stations. 



30. There is one case in which the deviation of the vertical may become unusually large, 

 which seems worthy of special consideration. 



For simplicity, suppose I to be constant for the land, and equal to zero for the sea, which 

 comes to regarding the land as of constant height, the sea as of uniform depth, and transferring 

 the defect of density of the sea with an opposite sign to the land. Apply the integral (40) to 

 those parts only of the earth's surface which are at no great distance from the station considered, 



so that we may put cos — = 1, sin — = i = — , if « be the distance of the element, measured along 

 •' '^ 2 2 2 2a 



a freat circle. In going from the station in the direction determined by the angle )^, suppose that 

 we pass from land to sea at distances «„ Sj, Sj,...and from sea to land at the intermediate distances 



5,^ 4^ On goino- in the opposite direction suppose that we pass from land to sea at the distances 



s-i, «_3, «-5, ..- and from sea to land at the distances «_,, s_,....Then we get from (40), 



JTT 



---- = a^/{logs, -log«_i - (logs, -log*.,) + log S3 -logs. 3- ...} cos^-rfx- 



0,1/ 



If the station be near the coast, one of the terms logSi, logs., will be large, and the zenith 

 will be sensibly displaced towards the sea by the irregular attraction. On account of the shelving 

 of the coast, the preceding expression, which has been formed on the supposition that I vanished 

 suddenly, would give too great a displacement ; but the object of this article is not to perform any 

 precise calculation, but merely to shew how the analysis indicates a case in which there would be 

 unusual disturbance. A cliff bounding a tract of table-land would have the same sort of effect as 

 a coast, and indeed the effect might be greater, on account of the more sudden variation of I. The 

 effect would be nearly the same at equal distances from the edge above and below, that distance 

 being supposed as great as a small multiple of the height of the cliff, in order to render the 

 expression (40) applicable without modification. 



31. Let us return now to the force of gravity, and leaving the consideration of the connexion 

 between the irregularities of gravity and the irregularities of the earth's coating, and of the 



