During the days when ships were powered by sail, the captains log documented everything that happended during the day. As sickness could spread rapidly on a ship,there were often times where the number of sailers that were ill exceeded the space provided in the log to record their names. During these times, the excess names of the sick were recorded in the next column, which was reserved for the weather conditions of the day. Thus, it was not unusal for an ill sailor to be listed "under the weather".
Don't get too close to Ahab, he is under the weather today.