GRAVITY AND GRAVITY POTENTIAL. I J 



gives the correction for other values of g, and the numerical values of these cor- 

 rections are given in tables 4H and 6h respectively of the Hydrographic Tables. 



14. Gravity Potential of Points at the Earth's Surface. According to the 

 modern principles of geodesy, levelings of high precision should always be combined 

 with determinations of the acceleration of gravity. This combination of leveling 

 with gravity measurements gives all the data required for the determination of 

 gravity potentials of points at the earth's surface. 



Leveling consists in sighting along level surfaces and in measurements of heights 

 normal to them. A curve consisting of successive horizontal and vertical parts is 

 thus traced out. Forming for this curve the integral (a), section 11, we have to take 



into account the vertical parts only. Let their lengths be z, z', z", , 



and let g, g' , g", be the mean values of the acceleration of gravity 



along each of them. The integral then takes the form 



{a) 4> t - ^ = gz + g'z' +g"z" + 



The sum on the right side thus gives the difference of gravity potential between the 

 end-points of the curve. 



All the measurements required for the determination of gravity potentials are 

 thus performed by modern geodetic work. But unfortunately the results are not 

 worked out and published in this form. Attention is directed mainly to the sum 



(b) Z=z + z' + z" + 



which is supposed to represent the difference of height between the two end-points. 

 This Z is, however, no well-defined quantity, because the level surfaces are not 

 parallel to each other. If the leveling be performed along another route, a slightly 

 different sum Z will generally be found. The discrepancies caused by the lack 

 of parallelism between the level surfaces may be diminished by suitable corrections, 

 but no general method can be conceived which would make them disappear, and 

 the real relation of the determined Z to the vertical distance of the one point from 

 the level surface passing through the other will remain obscure. 



The only quantity which can be determined without ambiguity is the gravity 

 potential <f>. The same will be the case if we pass to the other fundamental method 

 for the determination of heights, the barometric method. As we shall have occa- 

 sion to show later, this method also gives gravity potentials as its direct natural 

 result, while the passage to heights brings uncertainties. 



That under these circumstances gravity potentials, when wanted, must be found 

 by recalculation from the published heights, is very unsatisfactory, so much the 

 more so as it will probably presently become apparent that gravity potentials are 

 what are really needed for scientific purposes, heights being only of secondary 

 importance. Such at least is the case in meteorology, and will also be that of 

 geology as soon as the question of the statics and dynamics of the earth's crust is 

 taken up seriously. It would therefore be a great advance if gravity potentials were 

 published as the main scientific result of geodetic work, and heights only as results 

 computed from gravity potentials. 



