I 



THEORY OF THE MOTIOM OF ROCKET!. 1^5 



therefore* =: ( l^ b* X (hyp. log:. " ^ ) 4-^4 



\ \ ° am — cty 



o m x, N't 



When the angle of projection is go*, / z= 0, and A =r 1 : 



therefore v in this case will be 6 X hyp. loj^. — 



^ am — c t 



t g t ; as determined in Prop. 1 : and when A ir 0, or the 

 action of gravity is 0, the velocity of the rocket in its rectili- 

 near path is 6 X hyp. log. which agrees vrith what 



has already been observed. 



Prop. IV. 



To find the horizontal ravge of the rocket, having the 

 angle of elevation of the engine, and the time the rocket 

 is on fire given. 



Let D, fig. 2. be the place of the rocket when all the Horizontal 

 ^'ild-fire it contains is just exhausted ; and C Wjand C n range of a 

 the measures of the velocities of the rocket in the directions ^°^^^'* 

 A C and D I, the latter of which is a tangent to the curve 

 at D : then by trig. sin. Z C n ?« (= w C JB — I D B) z= 



Cm • ^ rf^ C »i 



77— X sm. Z C win z= - — X co-sin, of the angle of 

 C n L>n * 



V^l at C 

 elevation of the engine - ^.^^' ^^ ^ ^ co-sin. of the Z 



CAB. Whence calling the velocities at C atid D, V and 

 V (computed from the 3 Prop.) we have sin. Z I D B = 



V 



— y co-sm. Z C A B. Since, then, we have found the 



Z I D B, it will be easy to determine that part of the range 

 denoted by B L. For the curve from D being a parabola 



D H z= , and V E =: -~ — ^from the laws of proiec- 



tiles in vacuo) ; where 5 and u represent the sin. and co-sin. 

 of the Z I D H = Z i D B — 90" ; consequently V :f = 



VE + EFziVE + DBzz ll-!l + kx-^gt^ ; whereof 



X i« given by the first proposition. 



Again, 



