When a charged capacitor is connected to an uncharged capacitor charge flows from the charged to the uncharged capacitor until there is no net force on the charge carriers. This means that both capacitors are at the same potential, because if they were not, there would be an electric field
and hence a force on the charge carriers of magnitude
and the charge carriers would move to equalize the potentials and reduce the electric field and hence the force, eventually to zero.
Of course, as charge flows from one capacitor to the other, the total charge is conserved.
In the diagram above, when switch is closed chraged flows from the top capacitor to the bottom. When charge has stopped flowing,
Also, because the potentials are equal
(1)
Suppose then that the capacitor at top left is initially charged to potential of
and that it has capacitance
and is then connected to a capacitance
Initially
stores a charge
Then
(3)
From (1)
(4)
Sub (4) into (1) to get
Now use
for either capacitor to find
Using it for (2) gives