The geometry of the earth - Physical Geodesy - page 226

earth's surface

dH dn
d(OC)

local horizon

±'

P

parallel to geoid

plumb line

±'

geoid

ground

x (north)

P 0

geoid

parallel to
equatorial plane

Fig. 5.18. Plumb-line curvature and orthometric correction

According to Sect. 4.3, we have

H = C
g

dC = gdn = − dW , . (5-133)

Hence, (5-130) becomes

1

g dC = dH + 1

d (OC) = dH −

g dW ,

(5-134)

so that

δϕ = ∂H
∂x

∂W
∂x

+ 1
g

,

(5-135)

δλ cos ϕ = ∂H
∂y

+ 1
g

∂W
∂y

.

These equations relate the reduction for the curvature of the plumb line
to the orthometric height H and the potential W . In view of the irregular
shape of the plumb lines, it is remarkable that such simple general relations
as (5-131), (5-132), and (5-135) exist.