on the other hand, the difference of the vapor pressure at height h compared to the vapor pressure in the water at the base of the tube is approximately the negative of the weight of the vapor over a height h; i.e.,

Make this proper case.
On the other hand, the difference of the vapor pressure at height h compared to the vapor pressure in the water at the base of the tube is approximately the negative of the weight of the vapor over a height h; i.e.,