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9.

Such that [[underline]] m [[/underline]] is a minimum.

It is possible to put ^[[f(s)]] and ^[[df(s)/ds ds]] in place of [[underline] v [[/underline]] and [[underline]] dv [[/underline]], in equation (3), and to obtain [[underline]] m [[/underline]] by integration. But in order that [[underline]] m [[/underline]] shall be a minimum, ^[[ẟ]] [[underline]] m [[/underline]] must be put equal to zero, and the function ^[[f(s)]] determined. The procedure necessary for this determination presents a new and unsolved problem in the Calculus of Variations.

[[underline]] SOLUTION OF THE MINIMUM PROBLEM BY AN APPROXIMATE METHOD [[/underline]].

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, [[underline]] n [[/underline]], we may consider the quantities [[underline]] R [[/underline]], [[underline]] g [[/underline]], and also the acceleration, to be [[underline]] constant over each interval [[/underline]].

If we denote by [[underline]] a [[/underline]] 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 [[underline]] m [[/underline]] and [[underline]] t [[/underline]], as follows:

^[[dm/dt = (M-m)(a+g)+R/c(1-k),]]    

the solution of which, on multiplying and dividing the right number by (a+g), is

^[[M=e-a+g/c(1-k) + M(a+g)+R/a+g [∫e a+g/c(1-k) + (a+g/c(1-k)) dt + C]

= e -a+g/c(1-k) + M(a+g)+R/a+g [e a+g/c(1-k) t + C],]]

Transcription Notes:
The formulas involve exponents and it's not clear how to transcribe properly so that the transcription expresses the relationship. For example, the mathematical constant known as e is raised to a power that is expressed as a fraction. There are programs available on the internet for formatting complicated formulas, but you cannot paste results into the transcription window. There need to be some conventions for transcribing exponents.