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10.

where [[underline]] C [[/underline]] is an arbitrary constant.

This constant is at once determined as -1 from the fact that [[underline]] m [[/underline]] must equal zero when [[underline]] t [[/underline]] = 0.

We then have

^[[m = (M+ R/a+g) [1-e [[exponent]] - a+g/c(1-k) t [[/exponent]]  ].]]    (5)

This equation applies, of course, to each interval; [[underline]] R [[/underline]], [[underline]] g [[/underline]], and [[underline]] a [[/underline]], being considered constant. We may make a further simplification if, for each interval, we [[underline]] determine what initial mass, M, would be required when the final mass in the interval is one pound [[/underline]].  The initial mass at the beginning of the first interval, or what may be called the [[underline]"total initial mass" [[/underline]], required to propel the apparatus through the [[underline]] n [[/underline]] intervals will then be the [[underline]] product of the n quantities [[/underline]] obtained in this way.

If we thus place the final mass, (M-m), in any interval equal to unity, we have ^[[M=m+1]] and when this relation is used in equation (5), we have for the mass at the beginning of the interval in question 

^[[M = R/a+g (e [[exponent]] a+g/c(1-k) t[[/exponent]] -1) + e [[exponent]] a+g/c(1-k) t[[/exponent]] .]]     (6)

Now the initial mass that would be required to give one pound mass the same velocity at the end of the interval, if [[underline]] R [[/underline]] and [[underline]] g [[/underline]] had both been [[underline]] zero [[/underline]], is, from (6)

^[[M = e [[exponent]] at/c(1-k) [[/exponent]].]]     (7)

The ratio of equation (6) to equation (7) is a measure of the [[underline]] additional [[/underline]] mass that is required for overcoming the two resistances, [[underline]] R [[/underline]] and [[underline]] g [[/underline]]; and when this ratio is least, we know that [[underline]] M [[/underline]] is a minimum