OF MOTIOV CONFINKD TO GIVEN SURFACES. 



terminate in the same vertical line as the simple horizontal 

 motion; therefore the horizontal motion will remain un- 

 altered. 



252. Theorem. The greatest height to 

 which a projectile will rise may be deter- 

 mined by finding the height from which a 

 body must fall in order to gain a velocity 

 equal to its vertical velocity ; and the hori- 

 zontal range may be found by calculating 

 the distance described by its horizontal velo- 

 city in twice the time of rising to its greatest 

 height. 



This is evident from the equality of the velocity of 

 ascending and descending bodies at equal heights, and from 

 the independence of the vertical and horizontal motions of 

 the projectile. 



253- Theorem. With a given velocity, 

 the horizontal range is proportional to the 

 sine of twice the angle of elevation. 



The time of ascent being as the vertical velocity, or 

 the sine of the angle of elevation, the range is as the 

 product of the vertical and horizontal velocities, or as the 

 product of the sine and cosine ; that is, as the sine of twice 

 the angle (140). 



254. Theorem. The path of a projectile 

 moving without resistance, is a parabola. 



Since the horizontal velocity is 

 uniform, the times of describing 

 AB, AC, or X, are as their lengths, 

 and the spaces BD, CE;, describ- 

 ed by the accelerating force of 

 gravitation, as the squares of 

 these times, or as x-, whence 

 ^"^ay, and ADE is a parabo- 

 la, of which a is the parameter 



(204). 



Scholium. In practical cases the resistance of the 



atmosphere renders this theory of little use, except when 



the velocity is very small. 



SECT. V. OF MOTION CONFINED TO GIVEN 

 SURFACES. 



"255. Theorem. When a body descends 

 along an inclined plane, without friction, the 



VOL. II. 



B 



D 



force in the direction of the plane is to the 

 whole force of gravity as the height of the 

 plane is to its length. 



For if AB represent the motion which a 

 would be produced by gravity in a given 

 time, this may be resolved into AC and 

 CB (226) ; by means of AC the body ar- 

 rives at the line CB in the same time as if 

 it were at liberty ; but the motion CB is destroyed by the 

 resistance of the plane ; and as AB to AC so is AD to AB 

 (l2l). But forces are measured by the spaces described in 

 the same time (23o). 



256. Theorem. When bodies descend 

 on any inclined planes of equal height, their 

 times of descent are as the lengths of the 

 planes, and the final velocities are equal. 



(2Jr\ 1 



— I (233), and here azz—, «=^/(2Ix)=: 

 a / X 



v'2.r; and the times vary as the spaces, but the times 



being greater in the same proportions as the forces are less, 



the velocities acquired are equal (23o). 



257. Theorem. The times of falling 

 through all chords dr.awn to the lowest point 

 of a circle are equal. 



The accelerating force in any chord 

 AB is to that of gravity as AC to AB, or ]> 

 as AB to AD (l2l), therefore the fortes 

 being as the distances, the times are 

 equal ; for their squares are as the 

 spaces directly and the forces inversely 

 (233). 



258. Theorem. When abody is retain- 

 ed in any curve by its attachment to a thread, 

 or descends along any perfectly smooth sur- 

 face of continued curvature, its velocity is 

 the same, at the same height, as if it fell 

 freely. 



Since the velocity is the same at A, f. 

 whether the body has descended an 

 equal vertical distance from B or C, it 

 will proceed in AD with the same velo- ^ 



city in both cases, provided that no motion be lost in the 

 change of its direction, and therefore its velocity will be 

 the same after passing any number of surfaces as if it had 

 fallen perpendicularly from the same height. But where 

 F 



