NO. 2 METHOD OF REACHING EXTREME ALTITUDES 49 



to the /'.28th power ; and for bundles or groups of ship rockets, as 

 shown in figure 7, the minimum masses will be those for 7,000 

 ft./sec, raised to the By.Bth power. Even when air resistance is 

 entirely neglected in the calculations for the last case, the masses are 

 of much the same magnitude, as shown in table VII. The large 

 values of the masses Mri and Mr2 simply express the impossibility 

 of employing rockets of low efficiency. Attention may be called to 

 the particular case under Mr2 (the groups of ship rockets indicated 

 in fig. 7) in which one pound is raised to the altitude of 1,228,000 

 feet (232 miles) ; the " total initial mass " in this case, even neglect- 

 ing air resistance entirely, is 2.89 X 10^^ lbs., or over sixfold greater 

 than the entire mass of the earth. 



These large numbers, to be sure, agree with one's first impression 

 as to the probable initial mass of a rocket designed to reach extreme 

 altitudes ; but the comparatively small initial masses, possible with 

 high efficiency, are not intuitively evident until one realizes what an 

 enormous reduction is involved in extracting anything as large as 

 the 27th root of a number. 



It should be observed that the apparatus is taken as weighing one 

 pound. Strictly speaking, if the recording instruments have a mass 

 of one pound, the entire final mass of the apparatus must be at least 

 three or four pounds. The mass for the recording instruments may 

 be considered as being very small, yet many valuable researches 

 could, of course, be performed with an apparatus weighing no more 

 than this." The entire final apparatus should if possible be designed 

 to weigh not over 3 or 4 lbs. at most, unless the efficiency of the 

 apparatus is so high that the " effective velocity," c(i — k), is at least 

 in the neighborhood of 7,000 ft./sec. An examination of table VII 

 makes very evident the necessity of securing maximum effectiveness 

 of the apparatus before a rocket for such a purpose as meteorological 

 tvork, for example, is constructed; in order to make the method as 

 inexpensive as possible. It should be remarked, however, that the 

 " total initial mass " will really not be increased in as large a propor- 

 tion as the -final mass if the latter is made greater than one pound by 

 virtue of equation (2). 



Before proceeding further it will be well to consider carefully the 

 question of air resistance as dependent upon the cross-section of the 

 rocket during flight. It has already been assumed that the cross- 

 section, in the calculation of the minimum M for each interval, was 

 one square inch. If we make the apparatus as long, narrow, and 

 compact, as possible, the assumption of a cross-section of one square 



