50 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 71 



inch for an apparatus weighing one pound will not be unreasonable. 

 A glance at tables Y and VI will show that, for " effective velocities "' 

 of 7,000 ft./sec. and 3,500 ft./sec, the mass at the beginning of any 

 interval (except s^) does not greatly exceed one pound — the mass at 

 the end of each interval being one pound — so that the computations 

 are in agreement with this assumption of area of cross-section. For 

 the two cases of the adapted Coston rockets, the masses at the 

 beginning of the intervals are much larger ; and hence we see that 

 the " total initial masses " in table VII, large as they are, would have 

 been even larger if a proper value of cross-section had been employed. 



The important point is, however, that cross-sectional areas of 

 even less than one square inch should have been used. The reason 

 for this is obvious when one remembers that in calculating the " total 

 initial masses,'' when we multiply minimum masses, jM, to'gether we 

 are also multiplying the cross-sections in the same ratio. In other 

 words, we are considering numbers of rockets, each of one square 

 inch cross-section, grouped together side by side, into a bundle. But 

 such an arrangement would have its cross-section proportional to 

 its mass and not to the f d power of its mass, as would be the case if 

 the shape of the rocket apparatus ivere at all times similar to the 

 shape at the start (as in the ideal rocket, fig. i). This constant 

 similarity of shape is, as we have seen (equation 2), one of the condi- 

 tions for a minimum initial mass. Hence the " total initial masses " 

 that have been calculated are really larger than the true minima, 

 which would be obtained only by repeating the calculations, assuming 

 a smaller cross-section except in the last few intervals, in which the 

 rocket has become so small that the condition of one-square-inch-per- 

 pound is approximately satisfied. 



Before leaving the subject of air resistance, attention should be 

 called to the fact that the velocities (table V) do not exceed that for 

 which air resistance has been studied by Mallock until in s^, for 

 a=i50 ft./sec.^, and in Sg, for a = 50 ft./sec.-; and furthermore, that 

 the velocities do not become much in excess until the densities have 

 become almost negligible. 



CHECK ON APPROXIMATE METHOD OF CALCULATION 

 A simple calculation, involving only the most elementary formulae 



instead of equations (6) and (7) will show that the "total initial 



masses " in table VII cannot be far from the truth. 



Consider, for simplicity, a rocket of the form shown in figure i, 



and suppose that one-third of the mass of the rocket is fired down- 



