NO. 2 METHOD OF REACHING EXTREME ALTITUDES II 



for each interval, we determine what initial mass, M, would he re- 

 quired when the iinal mass in the interval is one pound. The initial 

 mass at the beginning of the first interval, or what may be called the 

 '"'' total initial mass," required to propel the apparatus through the n 

 intervals will then be the product of the n quantities obtained in this 

 way. 



If we thus place the final mass (M — m), in any interval equal to 

 unity, we have M = m+i and when this relation is used in equation 

 (5), we have for the mass at the beginning of the interval in question 



T? / a + g ^ \ a + g 



M= -^re^^^-i^) -ij+e'^^'-k> . (6) 



Now the initial mass that would be required to give the one pound 

 mass the same velocity at the end of the interval, if R and g had both 

 been sero, is, from (6) 



M = ecT^^k). (7) 



The ratio of equation (6) to equation (7) is a measure of the 

 additional mass that is required for overcoming the two resistances, 

 R and g ; and when this ratio is least, we know that M is a minimum 

 for the interval in question. The " total initial mass " required to 

 raise one pound to any desired altitude may thus be had as the product 

 of the minimum M's for each interval, obtained in this way. 



From equations (6) and (7) we see at once the importance of 

 high efficiency, if the " total initial mass " is to be reduced to a 

 ^minimum. Consider the exponent of e. The quantities a, g and t 

 depend upon the particular ascent that is to be made, whereas 

 c(i— k) depends entirely upon the efficiency o'f the rocket, c being 

 the velocit}^ of expulsion of the gases, and k, the fraction of the 

 entire mass that consists of loading and firing mechanism, and of 

 magazine. In order to see the importance of making c(i— k) as 

 large as possible, suppose that it were decreased tenfold. Then 



a+g ^ 



ec(i-k) would be raised to the loth poiver, in other words, the 

 mass for each interval would be the original value multiplied by 

 itself ten times. 



