GRAVITY AND GRAVITY POTENTIAL. 1 3 



10. Level Surfaces and Dynamic Height or Depth. A surface everywhere 

 perpendicular to the plumb-line is a level surface. The free surfaces of liquids in 

 equilibrium always form level surfaces, and the surface of the sea, together with its 

 continuation below the continents as referred to above, is the fundamental level 

 surface, to which all differences of level are referred. 



If gravity is the only acting force, no work is required to move a weight along 

 a level surface. But in order to lift it from one level surface to another, a certain 

 amount of work is required, and always the same amount, irrespective of where 

 on the two surfaces the two extreme points of the path are situated. Otherwise 

 perpetual motion could be realized by lifting the weight at the place where less 

 work is required and letting it down at the place where more work is required. 

 Any level surface is therefore specified without ambiguity by the amount of work 

 required to lift a certain mass, say unit-mass, from sea-level to any point of the sur- 

 face. Or, in other words, a level surface is a surface of equal gravity potential 

 (section 4) and is perfectly specified by the gravity potential of any of its points. 



The level surfaces must be carefully distinguished from the surfaces of equal 

 height above or equal depth below sea-level. The intensity of gravity decreases 

 from the pole to the equator. Consequently the unit-mass must be lifted higher at 

 the equator than at the pole, if the same amount of work is to be performed, and 

 thus the same level surface be attained. A surface of equal height above or of 

 equal depth below sea-level must therefore cut through the system of level surfaces. 

 The surface of equal height or depth is a slanting surface, which is not normal to 

 the plumb-line, and on which equilibrium is not possible under the sole action of 

 gravity. If the surfaces were hard and smooth a ball would remain in equilibrium 

 on a level surface. But on a surface of equal height above sea-level it would roll 

 in the direction from the pole to the equator; and on a surface of equal depth below 

 sea-level it would roll in the direction from the equator to the pole. 



This property at once shows that the surfaces of equal height or depth are not 

 suitable as coordinate surfaces in problems relating to the statics or the dynamics of 

 the atmosphere or the sea. For this purpose only level surfaces are found suitable. 



The introduction of the level surfaces as coordinate surfaces involves the use 

 of gravity potentials for the specification of heights and depths. With this applica- 

 tion of gravity potentials in view, we have introduced the names dynamic meter, 

 dynamic decimeter, etc., for units of this quantity. To standard values of the 

 gravity potential in the sense defined (section 4) will correspond standard equi- 

 potential surfaces. These will serve us as coordinate surfaces. 



We shall also use the expressions dynamic height and dynamic depth as synony- 

 mous with gravity potential, with the difference only that we take the dynamic 

 depth in the sea as a positive quantity, while the corresponding values of the 

 gravity potential are negative. By this mode of expression the level surfaces are 

 surfaces of equal dynamic height above or of equal dynamic depth below sea- 

 level, the height or depth of the standard surfaces being an integer number of 

 dynamic meters. We shall as a rule prefer the expressions dynamic height or 

 depth when we refer to the dynamic meter as unit, and the expression gravity 

 potential when we use the m.t.s. unit, the dynamic decimeter. 



