GRAVITY AND GRAVITY POTENTIAL. 19 



the geometric being only of secondary interest. But for maps on moderate scales 

 we can identify both kinds of topography. Near the coasts it will generally be 

 necessary to simplify the course of the curves. But for greater distances from the 

 coasts the bottom configuration is generally so regular, or our knowledge of it so 

 incomplete, that artificial simplifications may be more or less dispensed with. 



The topographical maps accompanying this work can be considered as repre- 

 senting both geometric and dynamic topography. On the map of the world, giving 

 the topography of the earth's surface both above and below sea-level, the main 

 curves are drawn for the interval of 1000 meters, which may be interpreted as 

 geometric or dynamic meters according to circumstances. For the displacement 

 from curve to curve of a unit-mass, we have a gain or loss of potential energy of 

 10,000 m.t.s. units. 



16. Scalar Field. It will be useful to refer here to some fundamental notions 

 relating to scalar fields, and their variations from place to place in space. Let a 

 be a scalar quantity which has a uniquely determined value in every point of space. 

 To represent distinctly the distribution in space of these values, or, in other words, 

 to represent the field of the scalar a, we can draw a set of equiscalar surfaces 



Each of these contains the points in space where the scalar has a certain constant 

 value, a , u a 2 , respectively. This is the well-known method of repre- 

 senting the distribution of potential by equipotential surfaces, that of pressure by 

 isobaric surfaces, that of temperature by isothermic surfaces, and so on. 



The sheet between two equiscalar surfaces a and a.^ will be called an equi- 

 scalar sheet. The use of the word " equiscalar " in connection with a sheet must 

 not be misunderstood. The scalar is not constant in the sheet, but it has limited 

 variations, the limits being given by its values <x and a x on the boundary. The 

 word "equiscalar" used for a sheet should remind us of this limitation of the 

 variations, as well as of the possibility of defining an average value of the scalar, 

 which is constant all along the sheet. 



In most cases it will be found convenient to draw the equiscalar surfaces for unit 

 differences of the scalar. These surfaces will then divide the space into a set of 

 equiscalar unit-sheets. Choosing a unit of suitable magnitude, we can always be 

 certain that the unit-sheets get a suitable thickness for a perspicuous distinct rep- 

 resentation of the field. If sufficiently thin sheets are obtained we can always say 

 that the difference between the values a x and a of the scalar in two points of space 

 1 and o is equal to the number of unit-sheets contained between them. 



This difference, a x a , divided by the length s of any curve joining the points 

 o and 1, 



gives the average rate of variation of the scalar along the curve s. 



