Gravity field of the earth - Physical Geodesy - page 43

equal within 10 − 11 , which is a formidable accuracy. He used the same type
of instrument by which experimental physicists have been able to determine
the numerical value of the gravitational constant G only to a poor accuracy
of about 10 − 4 , as we have seen at the beginning of this topic. The coinci-
dence between the inertial and the gravitational mass was far too good to be
a physical accident, but, within classical mechanics, it was an inexplicable
miracle. It was not before 1915 that Einstein made it one of the pillars of
the general theory of relativity!

2.2 Level surfaces and plumb lines

The surfaces

W ( x, y, z )=constant , (2-13)
on which the potential W is constant, are called equipotential surfaces or
level surfaces .
Differentiating the gravity potential W = W ( x, y, z ), we find

dW = ∂W
∂x

dx + ∂W
∂y

dy + ∂W
∂z

dz . (2-14)

In vector notation, using the scalar product, this reads

dW =grad W · d x = g · d x , (2-15)

where

d x =[ dx, dy, dz ] . (2-16)
If the vector d x is taken along the equipotential surface W = constant, then
the potential remains constant and dW = 0, so that (2-15) becomes

g · d x =0 .

(2-17)

If the scalar product of two vectors is zero, then these vectors are orthogonal
to each other. This equation therefore expresses the well-known fact that the
gravity vector is orthogonal to the equipotential surface passing through the
same point.
The surface of the oceans, after some slight idealization, is part of a
certain level surface. This particular equipotential surface was proposed as
the “mathematical figure of the earth” by C.F. Gauss, the “Prince of Mathe-
maticians”, and was later termed the geoid . This definition has proved highly
suitable, and the geoid is still frequently considered by many to be the fun-
damental surface of physical geodesy. The geoid is thus defined by

W = W 0 = constant . (2-18)