Consider a vertical section of a baroclinic fluid such as the ocean oriented parallel to the x-axis. The isobars and isopycnals in this vertical plane are defined by the equations
Thus the slopes of the isobars and isopycnals are
Under quasi-static conditions, p is related to 
through
the equation
Consequently, Eq. (42a) can be written in the form
Differentiating this equation with respect to z gives
The interchange of differentiation indicated above is permissible since p is a continuous function of x and z. Substituting from Eq. (43) gives
or taking g as constant for all practical purposes gives
The term 
remaining on the right
can be replaced by its equivalent from Eq. (42b) to obtain
Multiplying by dz and integrating from 
to 
gives
This can also be expressed in the form
where 
is the mean slope of the isopycnal
surfaces for the interval 
to 
. If the
isopycnals are constructed for equal increments of
then the value of can be estimated from the
arithmetic mean of the slopes of those isopycnals lying between
and .
Equation (49) can be written approximately as
where 
is the mean density for the layers
to . If is selected as that level at which
the isobaric slope is zero, then
This equation is the analogue of Eq. (40) for the case of a baroclinic fluid.
A relation identical to Eq. (50) or (51) applies for the slopes of the isobaric surfaces in the y direction. The magnitude of the combined slope is given by
and has the direction
relative to the x axis, where 
and
are the slopes determined from the relation (51) based on the
distribution of in the vertical planes oriented in the
x and y directions, respectively.