OF DEFLECTIVE FORCES. 129 



adx^=gdi^, we have d^^ — —dx", and fzix «/*"+/'; but if 



X, z, and t begin together, c=0, and/' = 0, consequentiy 



t :=: X js/ -» x:=: ^ —t, and zz=i cu;" + 6a:, whence zzn^a 

 9 « 



^-{■hts/ — :=.\qt'^-{-htJ — \ and these equations contain 



the whole theory of projectiles moving without resistance : 

 they show that the horizontal velocity is uniform, and that 

 the velocity in a vertical direction is the same as if the body 

 fell in a right line. 



[Scholium. It seems to be an unnecessary departure 

 from the simple order of investigation to examine a very 

 complicated and intricate case in order to deduce from it 

 a very simple one : and yet it may be said that unless this 

 were done, we should have frequent repetitions from con- 

 sidering the same case in its simple form, and then as an 

 inference from a more general law. But for a student, it 

 is better to have such repetitions, than to be without a 

 clear conception of the shortest path by which he may 

 arrive at an elementary conclusion. It seems, therefore, 

 not altogether superfluous to insert here a few illustrations 

 of the motions of projectiles, demonstrated in the most 

 natural and simple manner. 



274. Theorem. The velocity of a pro- 

 jectile may be resolved into two parts, its 

 horizontal and its vertical velocity : the hori- 

 zontal motion will not be affected by the 

 action of gravitation perpendicular to it, and 

 will therefore continue uniform ; and the ver- 

 tical motion will be the same as if it had no 

 horizontal motion. 



