**Motion of a Balloon Full at Starting**

The letters meaning the same as before, the equation of motion of a balloon completely filled at starting is

FORMULA

Or substituting for p and x their values

FORMULA

The integral of this differential equation could be obtained in series as before, only that the resulting equations would be more complicated. As we do not propose to discuss the formulae obtained, it will be sufficient for our purpose to deduce an approximate solution by neglecting Vo po (1 -- e --nz) compared to M, viz neglecting the mass of the gas that has escaped during the ascent compared to the mass of the whole balloon and appurtenances. It must be borne in mind, however, that when coal gas is used, and the ascent is to a great height, the mass of gas that escapes is by no means insensible. The equation thus becomes

FORMULA

x being FORMULA. This is an equation which can be integrated in exactly the same was as that previously considered, viz, by multiplying by a factor e-mx, and integrating at once; thus,

FORMULA

FORMULA

And C is determined as before by putting x = 0, when we have u = uo.

In this case uo is not zero, except when the balloon starts from the earth quite full. The general case is when the balloon is only partially filled on leaving; the previous equations then hold until a height h, at which it becomes quite full, when the motion changes, and is as just investigated. Then uo becomes the velocity at the height, h, and everything is measured from this height as if from the surface of the earth, a being then the radius of the earth + h, po, xo the densities at height, h,m and p, x at height x + h, &c. We have therefore, except as regards time, completely determined the motion of a balloon inflated with gas in an atmosphere of constant temperature. The introduction of temperature would modify the motion considerably, but in the present state of science it cannot be taken into account.

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