lO SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. Jl 



Evidently, then, the velocity of ascent must have some special value 

 at each point of the ascent. In other words it is necessary to deter- 

 mine an unknown function f (s), defined by 



v = f(s), 

 such that m IS a minimum. 



It is possible to put f (s) and ^^' ds in place of v and dv, in equa- 

 tion (3), and to obtain m by integration. But in order that m shall 

 be a minimum, 8m must be put equal to zero, and the function f (s) 

 determined. The procedure necessary for this determination pre- 

 sents a new and unsolved problem in the Calculus of Variations. 



SOLUTION OF THE MINIMUM PROBLEM BY AN APPROXIMATE 



METHOD 



In order to obtain a solution that will be sufficiently exact to show 

 the possibilities of the method, and will at the same time avoid the 

 difficulties involved in the employment of the rigorous method just 

 described, use may be made of the fact that if we divide the altitude 

 into a large number of parts, let us say, n, we may consider the 

 quantities R, g, and also the acceleration, to be constant over each 

 interval. 



If we denote by a the constant acceleration defined by v = at in any 

 interval, we shall have, in place of the equation of motion (3), a 

 linear equation of the first order in m and t, as follows : 



dm^ (M-m)(a-Fg)+R 

 dt c(i-k) 



the solution of which, on multiplying and dividing the right number 

 by(a + g),is 



-^^* _M(a + g)+R 



m: 



_e c(i-k) 



a + g 

 St^ M(a + g)+R 



•^-'•(c(^^))<"+^: 



a + l 



a+g 



gcd-k) +C 



where C is an arbitrary constant. 



This constant is at once determined as — i from the fact that m 

 must equal zero when t = o. 



We then have 



a +g t - 



m= (M-f- ~ - 

 ^ a + g 



I— e c(i-k) 



(5) 



This equation applies, of course, to each interval, R, g, and a. 

 being considered constant. We may make a further simplification if, 



