iggA THEORY OF THE MOTION OF ROCKETS. 



cif 

 zz t. hyp. log. [am — c t) -\- fluent of — /. hyp. 



am . . , , /- 



log. [am — c i) — t . hyp. log. [am — c /) — / /. 



) . hyp. log. [am — ct) — t zz [am — c t) , 



hyp. log, [am — ct') — t. So that the whole fluent will be 

 ar rr ft t . hyp. log. a m -\ [am — c/). hyp. log. (a m 



,., •,-,, , .ham 



— ct) -\- b t — g t which when x zz. o, and / n o is • 



hyp. log. a m. Hence the fluent corrected is j ::z: ( ft t 



)hyp. log. fl >7i + — (am — ct] , hyp. log. [am—^ 

 c 



e t) 4- b t • — §■*% and in the case where / z: a it is x zz 



(ahc — a b m\ ^ , . ab , » i , 

 j hyp. log, am ^ [m — cj . hyp. log. 



[a m —- a c) -\-ah~-'a^g~[c — m) . liyp. log. am + 



(m — c) . hyp. log. [am — ac) + c r- == "v'"* — *^^ 



(hyp. log. (a yn — <z e) — ' hyp. log. am\ + c ~ J tz 



, tt h /" , ,, , m — c. ftcg\ 



— X (^(m-c).hyp.log. -^ +c~.--^j. 



This in numbers is =: 127-0139 X (288 + hyp.log.T^^X 



160 — 1-133734) — 4015-9827734. So that it appears that 



by sunaming the foregoing series only to 6 terms gives the 



Heiglit to result within -376089 part of a unit, of this method. 



which the Since we have found the velocity at the end of this space to 



rocket would , "i ,, ■ ^ 



ascend if the be 2890*9 1 95 leet per second ; we shall , on the su pposition that 



retardation tj^g retaidivd force of oravity remains constant from D have 



from the force ^ "^ 



«.f gravity were c c p '» "" 2896-9895l« 



coiistant. by the theory or umiorm lorces „ r:' — : — ~ rn 



^ , 4^/ t)4 x-9993709 



131261-131 feet for the height to which the rocket wiU 

 farther ascend; which being added to that just determined 

 4015*9827735 ft. gives 135277*1137735 feet, for the whole- 

 height of the rocket above the surface of the Earth whea it 



has 



