7*0 



MECHANICAL PHILOSOPHY DYNAMIC* [cimurooAi KHWH. 



the range would be with the same charge of powder at 

 any other elev., r calling the maximum range, 



or that du ..-45, R, and the range 



due to u: ,) /, we have, from the 



result ju:" so tll:lt llT ho 'P of 



the maximum range, any other range, with the Sanaa 

 charge, may be easily found. ( >r if we know any range 

 r. eamawoding to "the .'.>v:i;-...:i ,.-. t!..n to MtnoiM 

 the range r', corresponding to another elevation a, we 

 have tho two equations, 



r* R sin. 2<i and r=R sin. 2a 



. f' sin. 2<i . sin. 2a f 



* r sin. - sin. 2a 



The following are the principal results in the foregoing 

 theory of projectiles in a non-resisting medium : 







Time of flight, t-2-sin. a 2 sin. 



Range, r -sin. 2a=2Asin. 2a, 

 9 



v* 

 Greatest range, R - 2A, 



Greatest height, H = fc sin.* a. 



Where g is 32 2 feet, a the angle of elevation, t the 

 velocity of projection, and h the height from which a 

 body must fall from rest to acquire that velocity. 



These results, however, ought to be regarded as but 

 purely theoretical. In practice, the resistance of the air 

 is so great in high velocities, as to render them almost 

 useless : the mathematical doctrine of projectiles, through 

 a resisting atmosphere, is full of difficulties ; and as the 

 laws of resistance are as yet but insufficiently established, 

 practical men must be guided chiefly by the results of 

 actual experiments. The experiments in gunnery, by 

 Dr. Button, as given in liis Mathematical Tract*, 

 may be consulted with advantage.* 



CIRCULAR MOTION CENTRIFUGAL FORCE. 

 If a body move uniformly in the circumference of a 

 circle, the foico to which that motion is due, must reside 

 in the centre of that circle. This is only a parti- 

 cular case of the following more general proposition : 

 namely, that if a body describe any curve, in virtue of a 

 force continually diverting it from its wonted rectilinear 

 path, that force must reside in a point such that, con- 

 ceiving a line to join that point with the moving body, 

 this line, moving with the body, must sweep over equal 

 sectional areas in equal times. 



This may be readily proved by first establishing the 



direct proposition, namely that if a body describe a 



about any centre of force, and a line be supposed 



to unite that centre with the body, equal areas will be 



described in equal times. 



Let an attractive force at S (Fig. 144), act upon a 

 body at the distance S B ; and to simplify the inquiry, 

 imagine at first, that instead of acting continuously, the 

 force exerts itself only at short regular 

 .la ; that, in fact, it consists of 

 [UJil successive impulses. 

 Let A B be tho rectilinear path de- 

 scribed by the body during any one 

 interval between two successive impulses. 

 At II the impulse is repeated, and the 

 course of the body is diverted into the 

 new path B C, which is described, during 

 the next interval ; at the end of winch 

 r impulse, iu the direction CS, 

 bends its path again, and so on. In tho 

 two intervals here considered, the tri- 

 angular areas SAB, SBC, will have 

 been described ; and it will be easy to 

 how that these areas, thus described in 

 equal times, are equal. For if, when 

 the body WM at B, no fresh impulse had been given to 

 aowrrrr, bar* Worn- of Httl* oat If appllrd tn calculating the 

 and balU of* oolcal akap*. aa oacd In the rtflr and 



Fig. 143. 



b r which Ite 



and windage" are 



it, it would have moved forward uniformly to M. 

 dMeribiof 15 M, th ooatbmliM of A B, equal to A B ; 

 in which case the triangular areas SAB, SB M, would, 

 of course, have been equal : it remains to show that the 

 latter of these is also equal to the triangular area 8 1 



The impulse from S, if it had acted alone on the Ixxly 

 when at B, would have brought it to some point X in 

 in the prescribed interval of time; and iU uniform 

 motion along AB, if undisturbed, would in the same 

 interval have brought it to M. In virtue of these two 

 uniform motions, the one from B to N, and the other 

 from B to M, in the same time, the actual motion of the 

 body is along the diagonal B C of the parallelogram M N. 

 II, -n.-e Ml' U-inic parallel to h S, tho twu trian-l.-.s 

 B M S, B C S, upon the same base B S, are between the 

 same parallels, and are therefore equal ; therefore tho 

 triangle B C S is equal to the triangle A B S. It follows, 

 therefore, that in the equal times, equal triangular 

 spaces have been described. 



In the same manner as it has now been shown, that 

 the triangular area S A B, 

 described in the first in- 

 terval of time, is equal to 

 that SBC described in 

 the second equal interval, 

 so may it be shown that 

 the triangular area de- 

 scribed in the third in- 

 terval, is equal to that 

 described in the second, 

 and so on. Hence these 

 triangular areas, in the 

 margin (Fis,'. 14o), being 

 all equal, it follows, uniting 

 any number of them in one 

 area, that equal areas are 

 thus described in equal 

 times. 



As this is true, however small the individual intervals 

 of time between the impulses may be, it is true when 

 these intervals are insensibly small, or when the suc- 

 cessive impulses follow in one continuous force, and the 

 lines AB, B C, etc., unite in one continuous curve. 

 Hence, when a body is constrained to move in a curve, 

 by the continuous action of a central force, equal areas 

 are described about the centre of force in equal times. 



Conversely, if equal areas are described about a point 

 in equal times, the centre of force governing tho motion 

 of the body must be at that point. 



Suppose the two equal areas SAB, SBC, to be de- 

 scribed about S in equal times (Fig. 144) : thru if tho 

 force acting on B were not in the direction BS, M C, 

 parallel to the direction of the force, would not bo pa- 

 rallel to B S ; that is, two triangles S H C, S B M on tho 

 same base B S, though not between the same parallels, 

 are equal, which is impossible (Euc. 40 of I). 



It follows from this, that, if a body under the influence 

 of a continuous force, move uniformly in tho circum- 

 ference of a circle, the force must be at the centre of 

 that circle, because equal sectional areas are described 

 iu equal times, since equal arcs are. 



The force by which a body moving in a curve is drawn 

 at any instant towards the centre of attraction B, is called 

 the coi'n'/"'"' force at the distance the liody is at that 

 instant : the opposing force with which tho body tends 

 to fly oil" from tho centre, and proceed in a m-tili 

 path, is called the etiitrifinjid force. In circular motion 

 tin-so forces are everywhere exactly equal ; for neither, 

 at any point of the circular path, prevails over tho other, 

 the distance from the centre of force S Wing even 

 the same. Let the arc A B (Fig. 140) be described in 

 one second of time: draw BE per]H-ndicular to A S ; 

 then in one second, the body, originally at A, will have 

 fallen from its wonted straight p.-i'li A Si, a distance = 

 A E towards the attractive force at S : hence, - \ ' 

 presses the intensity of that force (page ~'M) acting on 

 A. Join HA'; then since the arc A K diUers insensibly 

 from its chord (for the time of describing it m.-iv 1 

 garded as minute as we please), we may regard A B A' as 



