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45.

The altitude is divided into intervals short enough to justify the quantities involved in the above equations being taken as constants. The equations are then used to find the minimum value of [[underlined]] M [[/underlined]] for each interval, - the mean values of [[underlined]] R [[/underlined]] and [[underlined]] g [[/underlined]], in the interval, being employed - and the "total initial mass" required to raise a final mass of one pound to a desired altitude is then obtained as the product of these  [[underlined]] M' [[/underlined]]s.

[[underlined]] VALUES OF THE QUANTITIES OCCURING IN THE EQUATIONS [[/underlined]]. 

[[underlined]] The Effective Velocity, c(1-k) [[/underlined]].

The calculation which follows has been carried out with the assumption of a velocity of ejection of 7,500 ft/sec. and a constant, [[underlined]] k [[/underlined]], equal to 1/15. This velocity is considerably less than those that were actually obtained, both in air and [[underlined]] in vacuo [[/underlined]]. The "effective velocity" will thus be

c(1-k) = 7,000 ft/sec.

It should be noticed that [[underlined]] k [[/underlined]] could be 1/12 and yet not necessitate a larger velocity of ejection than 7,640, which is also under the highest velocities obtained in the experiments. It is important at this point to remember that the velocities [[underlined]] in vacuo [[/underlined]] would doubtless have been found to be considerably higher than the above value; in friction could have been eliminated in the "direct-lift" method.

[[underlined]]The Quantity, R [[/underlined]].

The mean value of [[underlined]] R [[/underlined]] for any interval was most easily obtained from a graphical representation of [[underlined]] R [[/underlined]] as a function of [[underlined]] v [[/underlined]]; the mean value of [[underlined]] R [[/underlined]] between the beginning and end of the interval being taken. Three curves were used for this purpose; for velocities