J34 THEORY OF THE MOTION OF ROCKEIS. 



Earth's centre, the rocket will ascend nearly 746.54121 feet 

 higher from a point 4230'609 feet above the Earth's surface 

 with a velocity of 28y6*9895 feet per second, than it v^ould 

 do if the same force as at the point D had continued con- 

 stant, or had continued to act upon the body always with 

 Force that *''*^ same intensity. Hence also, if the rocket had a velo- 

 would prevent city of 289()*9895 feet per second upwards when at a height 

 its returjiing. 4 <>• r^ 



from the Earth's surface = -^4 ^» 't would never 



c 



return ; but continue to move for ever, or fly off to an 



4 2" r* 



infinite diftance. For the expreffion for x is . -^ 



4gr — ac 



4 <i T* 



or X rr 5"^^ ;; where it is evident that on ac~ be- 



4gr — ac 



coming — 4g- r*, x will be infinite ; and therefore to find c, 

 we have only to put 4 §• r^ — a c* zz o and reduce the equa- 

 tion. 



Whence, having the height from which the body rauft 

 fall to acquire a velocity, which, being added to that of 

 2896*9895 feet per second, shall cause it to move perpe- 

 tually when projected with the velocity of their sum ; we 

 can readily determine what that velocity is ; and it being a 

 very curious fact to know, we will therefore give a solution 

 to the problem in this place. 



Put d ■zz -^—r- — C I the eiven heitrht from the cen- 



tre C 

 a: — CD, any variable height from tlie same 



point greater than the rad. C L 

 r-CL 



r 

 Then —^ is the accelerative force of gravity at D when that 



at the^urface is 1. Therefore^ izz — 2 gfx ; and the fluent 



4 e* r* 

 of the same is v^ — -^^ — •; which when properly corrected is 



»* - Agr^f- ^ -^ — (whenx — r)4gr^ xf — ^ 



- 4gr' 



