P₂ = Maximum system design operating pressure, psia.
V₂ = The compressed volume of gas at maximum
system’s pressure P₂ , cu. in.
P₃ = The minimum system pressure, psia, at which
the additional volume V₃ of lube oil is required.
V₃ = The expanded volume of gas at minimum system pressure P₃, cu. in.
Based on a Boyle’s Law for the following states:
P₁ V₁ⁿ = P₂ V₂ⁿ = P₃ V₃ⁿ
or
V₁ (P₁)^{1/ n} = V₂ (P₂)^{1/ n} = V₃ (P₃)^{1/ n}
and
V_x= V₃ - V₂,

From Boyle’s Law for the changing conditions from
the fully charged accumulator to the discharge state:
V₂ (P₂)^{1/ n} = V₃ (P₃)^{1/ n}
and
V₃= V_x + V₂,
or
V₂ (P₂)^{1/ n} = (P₃)^{1/ n}(V_x + V₂),

and finally with respect to V₂:

V₂ =              (P₃)^{1/ n} V_x
            -------------------------------
                   (P₂)^{1/ n} - (P₃)^{1/ n}

From Boyle’s Law for the changing conditions from
the pre-charged state to the fully charged accumulator:
(P₁)^{1/ n} V₁ = (P₂)^{1/ n} V₂

or