Let P_1 denote the pressure at the surface of the mercury inside a...

In a mercury barometer, the space above the mercury column inside the closed tube is a vacuum (known as Torricellian vacuum). Therefore, the pressure at the surface of the mercury inside the tube is zero, giving $P_1 = 0$.

The surface of the mercury in the cup is open and exposed to the atmosphere. Therefore, the pressure at this surface is equal to the atmospheric pressure, giving $P_2 =$ atmospheric pressure.