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49.

The mean densities in intervals [[underline]] s [[/underline]] [[subscript]] 1 [[/subscript]] to [[underline]] s [[/underline]] [[subscript]] 6 [[/subscript]] inclusive were obtained from Fig. 21, on which these intervals are marked. The remaining densities were estimated as already explained.
 
[[underline]] CALCULATION OF MINIMUM MASS FOR EACH INTERVAL [[//underline]].

The Tables V and VI are calculated for a start, respectively, from sea-level and from an altitude of 15,000 ft.; i.e., the beginning of [[underline]] s [[/underline]] [[subscript]] 3 [[/subscript]]. The procedure in each case is, however, identical. 

The process of calculation is as follow: At the beginning of any interval we have the velocity already acquired during the previous intervals, let us say [[underline]] v [[/underline]] [[subscript]] 0 [[/subscript]]. This velocity is, of course, zero at the beginning of the first interval. Assume any final velocity at random [[underline]] v [[/underline]] [[subscript]] 1 [[/subscript]], for the interval in question. 

The value of [[underline]] a t [[/underline]] may be had from the equation

v [[subscript]] 1 [[/subscript]] = v [[subscript]] 0 [[/subscript]] + a t, (9)
and [[underline]] t [[/underline]] is at once obtained from the relation

^[[s = V[[subscript]] 0 [[/subscript]]t + 1/2 at [[superscript]] 2 [[/superscript]]; i.e., 
t = s/v[[subscript]] 0 [[/subscript]] + 1/2 at, (10)]]
whence, of course, [[underline]] a [[/underline]] is at once known. 

The calculation of ^e a+g/c(1-k) [[superscript]] t [[/superscript]] and e at/c(1-k) call for no comment; and [[underline]] R [[/underline]] is obtained as [[/underline]] P [[/underline]], the mean ordinate between [[underline]] v [[/underline]] [[subscript]] o [[/subscript]] and [[underline]] v [[/underline]] [[subscript]] 1 [[/subscript]] from the curves as already explained, multiplied by [[underline]] S [[/underline]] and ^[[S/S[[subscript]] 0 [[/subscript]]]].

The value of [[underline]] M [[/underline]], the initial mass, for the interval, necessary in order that the final mass in the interval shall be one pound, is the obtained from equation (7); and finally, the ratio of equations(8) to (7) (i.e., ^[[M/e at/c(1-k)]]) is calculated. This is the ratio of the initial mass necessary, including losses due to both [[underline]] R [[/underline]] and [[underline]] g [[/underline]], to the mass necessary to give the one pound the same velocity, [[underline]] v [[/underline]] [[subscript]] 1 [[/subscript]],