We now ready to take the In Toto approach to the equilibrium of the milk in the bottle.
Applying In Toto to the Milk Bottle Problem,
we consider the total volume of milk in total,
and consider the equilibrium of the milk (before and after separation).
Both before and after, there is a gravitational force W acting downwards on the milk,
equal to the (unchanged) total weight.
This is balanced by the sum of all the reaction forces acting the milk.
Now the sum of the horizontal components of the sidewall forces must be zero --
and obviously these horizontal components cancel each other because of the radial symmetry
of a milk bottle.
But clearly, along the sloping sides of the bottle, the vertical components on opposite sides
of the bottle add together, to produce a net downward force which we call X, in the case of homogenous milk,
and X' in the case of separated.