Jg^ THEORY or THE MOTION OF ROCKETS. 



and thqs finding several point? of the curve, tbc curt« 

 itself may be d escribed . 



We have here supposed, gravity to act in parallel lines, 

 wl^ich 16 not strictly true ; but the distance to which a 

 xoci^et ranges on the earth's surface being 8o very amall 

 compared with its circumference, the errour arising from the 

 contrary supposition will iu>t iu any material degree affect 

 <^ur conqlusiono. 



pnop. in. 



Tq find the velocity of th^ rocket in the curve at any given 

 instant. 



Velocity of a In the preceding diagranv let A C n x, and A D rz r being 



rocket at any ^he space described by the rocket in the time t : Then call- 

 giYcu time. 



ing the velocity at C f— 6 X hyp. log. (Prop. 1.) 



V ; the velocity at D, in the curve, will be expressed genie- 



rally by ~t— , following from the laws for the resolution of 



motion. Now by the laws of falling bodies in vacuo C D 

 r: g" I* : and putting A: and / for the natural sine and co-sine 

 (to rad. 1.) of the angle C A B of projection ; we shall 

 have A B =: / x, C B n A: x, and D B (the ordinate of 



the curve) -zzk x^-g t"^. Therefore k zz ((Jc x — 1g t f)* 



+1-^ and M= y^ = ('lil±iii^l£££il)i 

 X V. Again by the theory of variable motions x 

 zz V / . Consequently r — I ! — - — : -2 '- \ 



X V = (^V* /* + (/b V ^ <2g <*)y - Q^h* X (hyp. log. 



— ^— y + (/t 6 X hyp. log. —^ 2 ^ ^'iV 



the velocity of the rocket at D ; which wants no correction : 

 because whtn v n o, / z: o, and the whole vanishes ■. 



tkerefore 



