NO. 2 METHOD OF REACHING EXTREME ALTITUDES 6/ 



(or groups), provided, as is, of course, to be expected from dimensional con- 

 siderations, that the larger any individual rocket, the less, in proportion, need 

 be the ratio of weight of metal to weight of propellant. 



Such a calculation can be made by finding the number of secondary rockets, 

 for case (b), that would be required for the same total initial mass, other 

 conditions being the same, as for continuous loss of mass with zero relative 

 velocity, which is practically case (a). 



For the latter, equation (7), in which R and g are neglected, is evidently 

 sufficient for the purpose, for the reason that the form of the expression, so 

 far as (i-k) is concerned, is the same whether or not R and g is included. 



Let us now find what conditions must hold for case (b), in order that the 

 total initial mass shall equal that for case (a). Assume, first, that the casings 

 are discarded successively at the end of n equal intervals of time, no mass 

 being discarded except at these times ; the velocity of gas ejection being c, 

 as before. The total initial mass is obtained as the product of the initial 

 masses for each interval, from equation (7) with k=io, assuming the final 

 mass for each interval is, as before, i lb., after first multiplying the initial 

 masses by a greater factor than unity, the excess over unity being the weight h, 

 of the casings which are discarded at the end of the intervals. 



If, in case (a), we divide the time into n equal intervals in the same way, 

 we shall have, as the condition that the total initial masses are the same in the 

 two cases. 



a(t/n)n a(t/n)n 



M = e'=^'~''- =(i+h)ne " . (15) 



We obtain, then, on combining (15) with (7), 



Mk= (I +h)n, 

 from which 



, log M 



log(i+h) 



(16) 



Let us assume, for case (a) (many small secondary rockets), as well as for 

 case (b) (large secondary rockets), that the ratio of mass of metal to mass 

 .of propellant is the minimum reasonable amount that can be expected, which 

 may be put tentatively, at least, as 1/14 and 1/18, respectively. 



Two cases will suffice for purpose of illustration : one in which the ratio of 

 initial to final mass is moderately large, e. g.. 40, and the other in which the 

 ratio is extreme, e. g., 600. 



The numbers of secondaries (or separate groups) for (b), for these two 

 cases, are, from (16), 5 and 9 respectively, n being necessarily an integer. 



It is to be understood that the numbers could be made even smaller, although 

 this would necessitate larger total initial masses. 



'' If the start were made at a greater elevation than 15,000 feet, for example, 

 at 20,000 or 25,000 feet, the reduction of the " total initial mass " would, of 

 course, be considerably greater. Further, if the rocket were of comparatively 

 small mass, it could be raised to an even greater initial height by balloons. 



*^ Actually, 300 grams would be sufficient, for many researches. 



^^ Attention is called to the fact that hydrogen and oxygen, coml^ning in 

 atomic proportions, afford the greatest heat per unit mass of all chemical 

 transformations. For this reason, if the calculations are made under the 



