2 2 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



equal to the difference of the values which the corresponding scalar quantity has in 

 the end-points of s, divided by the length of s. 



18. The Gravitational Field of Force. The relation of gravity potential to the 

 acceleration of gravity is that of a scalar quantity to its gradient. The gravity 

 potential will therefore serve to give us not only a rational system of coordinates ; 

 it will also give us a full representation of the gravitational field of force. 



To sum up the facts relating to this representation, we see that formula (a), 

 section 1 1, which defined the gravity potential in terms of the acceleration of 

 gravity, has exactly the form of the formula (e), section 17, which defines a scalar 

 quantity in terms of the gradient. Vice versa, the acceleration of gravity can be 

 represented by the rate of decrease of the gravity potential along the normal to the 

 equipotential surfaces, i. e., along the plumb-line z, 



In the same way the component of the acceleration of gravity along any direction 

 5 is given by the rate of decrease of the gravity potential along this direction 



Another form of expressing the facts contained in the formulae (a) and (&) is the 

 statement that the equipotential surfaces and the unit-sheets give a full representa- 

 tion of the gravitational field of force. First, the acceleration of gravity is directed 

 along the normal to these surfaces, i. e., along the plumb-line. Second, it is 

 numerically equal to the reciprocal thickness of the unit-sheets. Thirdly, its com- 

 ponent along any line 5 is numerically equal to the reciprocal length of the segment 

 of this line which is contained in the unit-sheet. As we see, these statements are 

 simply the reversal of the statements by which we defined originally our unit ot 

 gravity potential, the dynamic decimeter, in terms of the acceleration of gravity. 

 Corresponding to (f), section 17, we get finally 



(') -, = ^- 



where g- sm is the average value of the component which the acceleration of gravity 

 has tangentially to the curves, while <f> and 4>i are the values of the gravity potential 

 in the end-points of the curve. 



The gravitational field of force is a field in space and thus a 3-dimensional field. 

 The components of its field intensity tangentially to a surface will represent a 2- 

 dimensional field of force. These 2-dimensional fields, which will be of great 

 importance for us, are represented fully by a map giving the dynamical topography 

 of the surface. 



We can exemplify this by reference to our maps of dynamic topography. 

 Formula (c) can be used to find the average value of the acceleration of gravity 

 along any part of a curve contained in the surface represented by the map. Any 

 such curve will be divided into segments s by the level curves of the maps, and to 

 each such segment the formula (c) can be brought into application. 



