There are several factors to complicate any calculations. Any expansion of the gas bag--stretching due to an increase in internal pressure--would be one; but the envelope fabrics do not stretch much; there is indeed a very good reason why they must not be allowed to stretch. The pressure in the gas bag is a factor. If there is no stretching of the bag, this pressure will vary directly with the temperature of the gas, and might easily become excessive when the sun shines on the envelope. A more serious matter is the increased difference between the internal pressure of the gas and the external pressure of the atmosphere at high altitudes. Atmospheric pressure decreases as we ascend. The difference between gas pressure and air pressure thus increases, and it is this difference of pressure which tends to burst the envelope. Suppose the difference of pressure at sea level to have been two-tenths of a pound. For a balloon of twenty feet diameter, this would give a stress on the fabric, per lineal inch, of twenty-four pounds. At an altitude of 2000 feet, the atmospheric pressure would decrease by one pound, the difference of pressures would become one and two-tenths pounds, and the stress on the fabric would be 144 pounds per lineal inch--an absolutely unpermissible strain. There is only one remedy: to allow some of the gas to escape through the safety valve; and this will decrease our altitude. Ascending and Descending To ascend, then, we must discard ballast: and we cannot ascend beyond a certain limit on account of the limit of allowable pressure on the envelope fabric. To again descend, we must discharge some of the gas which gives us lifting power. Every change of altitude thus involves a loss either of gas or of ballast. Our vertical field of control may then be represented by a series of oscillations of gradually decreasing magnitude until finally all power to ascend is gone. And even this situation, serious as it is, is made worse by the gradual but steady leakage of gas through the envelope fabric. Here, in a word, is the whole problem of altitude regulation. Air has no surface of equilibrium like water. Some device supplementary to ballast and the safety valve is absolutely necessary for practicable flight in any balloon not staked to the ground. A writer of romance has equipped his aeronautic heroes with a complete gas-generating plant so that all losses might be made up; and in addition, heating arrangements were provided so that when the gas supply had been partially expended its lifting power could be augmented by warming it so as to decrease its density below even the normal. There might be something to say in favor of this latter device, if used in connection with a collapsible gas envelope.