686 Mn. STOKES, ON THE VARIATION OF GRAVITY 



If we cared to leave the mean value of gravity unaltered, we should have to use, instead of ^, 

 its excess over its mean value S^. In considering, however, only the variation of gravity from one 

 place to another, this is a point of no consequence. 



21. In order to estimate the magnitude which the quantity 3g" is likely to attain, conceive 

 two stations, of which the first is surrounded by land, and the second by sea, to the distance of 

 1000 miles, the distribution of land and sea beyond that distance being on the average the same at 

 the two stations. Then, by hypothesis, the potential due to the land and sea at a distance greater 

 than 1000 miles is the same at the two stations; and as we only care for the difference between the 

 values of the potential of the earth's coating at the two stations, we may transfer the potential due 

 to the defect of density at the second station with an opposite sign to the first station. We shall 

 thus have around the first station, taking h' for the depth of the sea around the second station, 

 S =ah + {a - 1) /i. In finding the difference V of the potentials of the coating, it will be amply 

 sufficient to regard the attracting matter as spread over a plane disk, with a radius s equal to 1000 

 miles. On this supposition we get 



Now G = t Trpo, and therefore 3g- = — = ; G = -. . - G. Making the same 



3 " ' 2a ipa 4 pa a 



suppositions as before with regard to the numerical values of a, p, h, h', and a, we get 



3g" = -000147 G. This corresponds to a difference of 6-35 vibrations a day in a seconds' pendulum. 



Now a circle with a radius of 1000 miles looks but small on a map of the world, so that we may 



readily conceive that the difference depending on this cause between the number of vibrations 



observed at two stations might amount to 15 or 20, that is 7.5 or 10 on each side of the mean, or 



even more if the height of the land or the depth of the sea be under-estimated. This difference 



will however be much reduced by using kg" in place of 3g"*. 



22. The value of V^ at any station is expressed by a double integral, which is known if S be 

 known, and which may be calculated numerically with sufficient accuracy by dividing the surface 

 into small portions and performing a summation. Theoretically speaking, T, could be expressed 

 for the whole surface at once by means of a series of Laplace's coefficients ; the constants in this 

 series could be determined by integration, or at least the approximate integration obtained by 

 summation, and then the value of V^ could be obtained by substituting in the series the latitude and 

 longitude of the given station for the general latitude and longitude. Rut the number of terms 

 which would have to be retained in order to represent with tolerable accuracy the actual state of the 

 earth's surface would be so great that the method, I apprehend, would be practically useless ; 

 although the leading terms of the series would represent the effect of the actual distribution of land 

 and sea in its broad features. It seems better to form directly the expression for F, at any station. 

 This expression may be calculated numerically for each station by using the value of S most likely 

 to be correct, if the result be thought worth the trouble; but even if it be not calculated 

 numerically, it will enable us to form a good estimation of the variation of the quantity 3g" or kg" 

 from one place to another. 



Let the surface be referred to polar co-ordinates originating at the centre, and let the angles 

 \1/, y be with reference to the station considered what 6, <p were with reference to the north pole. 

 The mass of a superficial element is equal to Sa- sin v|/dx|/dj^, and its distance from the station is 



2o sin — . Hence we have 



F, = fl//5cos|^dx|/dx (31) 



• The efFectof the irregularity of the earth's surface is greaier than what is represented by kg", for a reason which will be explained 

 further on (Art. 25). 



