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8.

cross-section is greatest at the head of the apparatus, as shown in United States Patent No. 1,102,653.

The quantities [[underline]] R [[/underline]], [[underline]] g [[/underline]], and [[underline]] v [[/underline]], are evidently expressible most simply in terms of the altitude [[underline]] s [[/underline]], provided the cross-section [[underline]] S [[/underline]] is also so expressed, giving, in place of equation (1)

^[[C(1-K)dm = (M-m)dv + 1/v(s) [R(s) + g(s) (M-m)]ds.]] (3)

[[underline]] RIGOROUS SOLUTION FOR MINIMUM M AT PRESENT IMPOSSIBLE [[/underline]].

The success of the method depends entirely upon the possibility of using an initial mass, [[underline]] M [[/underline]] of explosive material that is not impracticably large.  It amounts to the same thing, of course, if we say that the mass ejected up to the time [[underline]] t [[/underline]]  (i.e., [[underline]] m [[/underline]]) must be a minimum; conditions for the existence of a minimum being involved in the integration of the equation of motion.

That a minimum mass, [[underline]] m [[/underline]], exists when a required mass is to be given an assigned upward velocity at a given altitude is evident intuitively from the following consideration: If, at any intermediate altitude, the velocity of ascent be very great, the air resistance [[underline]] R [[/underline]] (depending upon the square of the velocity) will also be great.  On the other hand, if the velocity of ascent be very small, force will be required to overcome gravity for a long period of time.  In both cases, the mass necessary to be expelled will be excessively large.

Evidently, then, the velocity of ascent must have some special value at each point of the ascent.  In other words, it is necessary to determine an unknown function ^[[f(s)]], defined by

^[[v=f(s),]]