Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P-V diagram. The relation between the ratio $$\frac{V_a}{V_d}$$ and the ratio $$\frac{V_b}{V_c}$$ is:

For an adiabatic process,

$$PV^{\gamma}=\text{constant}$$

There are two adiabatic curves:

For adiabatic through a and c:

$$P_aV_a^{\gamma}=P_cV_c^{\gamma}$$

So,

$$\frac{P_a}{P_c}=\frac{V_c^{\gamma}}{V_a^{\gamma}}$$

For adiabatic through b and d:

$$P_bV_b^{\gamma}=P_dV_d^{\gamma}$$

So,

$$\frac{P_b}{P_d}=\frac{V_d^{\gamma}}{V_b^{\gamma}}$$

Now a and b lie on same isotherm, so

$$P_aV_a=P_bV_b$$

$$\frac{P_a}{P_b}=\frac{V_b}{V_a}$$

Also c and d lie on same isotherm, so

$$P_cV_c=P_dV_d$$

$$\frac{P_c}{P_d}=\frac{V_d}{V_c}$$

Now divide the two adiabatic equations:

$$\frac{P_a/P_c}{P_b/P_d}=\left(\frac{\frac{V_C}{V_a}}{\frac{V_d}{V_b}}\right)^{\gamma}$$

Using isothermal relations,

$$\frac{(V_c/V_a)}{(V_d/V_b)}=\left(\frac{V_bV_c}{V_aV_d}\right)^{\gamma}$$

This simplifies only if

$$V_bV_c=V_aV_d$$

Hence,

$$\frac{V_a}{V_d}=\frac{V_b}{V_c}$$