739 



PROJECTILES, THEORY OF. 



PROJECTION. 



780 



by considering that x = OVhen t 0; whence coiiat. hyp. log. cos. 



c-gt 



o u 2 , cos, u. 



-: therefore x = hyp. log. 



COS. p 



Making v in the above equation for v, we get the value of * 

 when the body has attained its greatest height ; and substituting the 

 value of t so found in the last equation for x, we have that greatest 

 height. 



When arrived at the greatest height, the body would begin to return 

 towards the earth ; and it may be shown that the velocity acquired by 

 the body on arriving at the place from whence it was projected would 

 be less than the initial velocity; also that the time of the descent 

 would differ from the time of ascent. 



If we imagine the earth to be perforated in the direction of a 

 diameter ; and if a body be allowed to descend towards the centre in a 

 non-resisting medium from any point in the line of perforation : the 

 law of attraction being in such a case directly proportional to the 

 distance of the body at any time from the centre of the earth (Newton, 

 Eb. i. prop. 73), the relation between the space descended and the time 

 of descent may be thus investigated. 



Let r be the radius of the earth, and let the force of gravity at the 

 surface be represented by g; then x being any distance from the 

 centre, the attracting force acting on the body at that distance will 



gx 

 be . Therefore since the distance of the body from the centre 



diminishes, while the time reckoned from the moment of departure 

 increases, we shall have -JJTJ = . This equation will be found to 



be verified by assuming x = re cos. t V~ + & sin- ' V~ ; which being dif- 



dx n fj g g 



ferentiated once, gives "77 = o Vj sm - ' V~ + & V7 c s - ' Vr- Now, 



in the equation for x, making x=r'(any given distance from the 



dx 

 centre) when t= 0, we have ar" ; and in the equation for -77, making 



dx 



^7 (the velocity) =0 when < = 0, we have i=0. Consequently x=r' 



cos.<V~; whence x is found when t is given: but when x=0, we 



have t A/~ = jj (where ir represents the half circumference of a 

 circle whose radius is equal to unity) whatever be the value of /. 

 Therefore t= jj V/ will express the time of falling from the surface 



to the centre of the earth. 



Let it now be required to investigate the relations between the 

 times, the spaces described, and the acquired velocities when a body 

 falls in vacuo from a point at such a distance from the earth that the 

 attraction of gravity upon it may be considered as variable ; and when, 

 agreeably to the law of nature, its intensity Is inversely proportional 

 to the square of the distance. (' Princip.,' lib. i., prop. 74.) Then, if 

 r be the radius of the earth, p the distance from the centre of the 

 earth to the point above the latter from whence the body is let fall ; 

 and if x be the space descended in any time t : also if g be the force 



1 1 rg 



of gravity at the earth's surface, we shall have -5 : g : : , _ ., : -. _ -r, ; 



and the last term expresses the force of gravity at the place of the 

 body when the space descended is x and the time of descent is t : 



therefore -77? = ; :;. 



Oi \P~~ Xf 



In order to integrate this equation, multiply both sides of it by Zdx ; 



da? Ztfr 3 

 and then the first integral will be -77$ = ~~ + contt. The constant 



may be found on considering that -77 (the velocity) =0 when < = 0, 



when also x 



dx 

 whence -57 = 



dt 

 ; therefore contt. = , and 



di? 





2</r* x 1 



~T~ r . This equation may be put in the 



-r- dt; and by the rules of integration we have 



~~~ 



+ "2 P arc cos - = 



p 1 



there is no constant 



n 

 to be added, because x=0 when t=0. From this equation t may 



easily be found when x is given : likewise from the equation for ^7 



we have the velocity when x is given. And if x be made equal to 

 p r, the whole distance of the body from the surface of the earth, we 

 shall obtain the whole time of the descent and the velocity acquired at 

 the end of that time, 



Again :_ let it be supposed that a body may lie projected vertically 

 upwards in vacuo from the surface of the earth, and be subject to a 

 variably accelerative force of attraction downwards. Let ) be the 

 semidiameter of the earth as before; and now let x be the height 

 ascended from the surface at the end of the time t : also let h be the 

 height due to the initial velocity, supposing the latter to have been 

 acquired by a body falling in vacuo with a uniformly oceelerative 

 force ; then Igh will express the square of that velocity. By the law 



of attraction we have 55 : g : : (r+x ^ : frf^u ! and the last term ex- 

 presses the intensity of the attractive force at the end of the time t 



from the commencement of the ascent. Hence -;-; = -~^- 



dt 1 ' 



In order to integrate this equation, multiply both sides by 2dx 



then we get 



+ const. : and to find the constant, it must 

 dx 3 



be observed that a;=0 when -^ = 2gk; therefore const. = 



dx" 

 and. ~TTZ = 



2.7^ 



Taking the square 



roots and transposing, we have ^/T 

 this equation may be put in the form 



(ft + ,. )iY . 



dx = 



, dt, and 



or (A being small compared with r) rejecting k when added to r, the 

 equation becomes 



Now, multiplying the numerator and denominator of the first member 

 by 2r, and putting the whole in the form 



(r* + 2rx)dx 



r dx 



the rules of integration give 



V{fo-* + ** + ;'} ]+ const. V3~ t. 



The constant may be determined by considering that x=0 when 10', 

 and thus t may be found when x is given. 



\Vhat has been stated respecting the vertical descent and ascent of 

 bodies may be understood to apply also to bodies descending and 

 ascending on inclined planes ; the force of gravity on the plane being 

 represented by g sin. a, where o is the inclination of the plane to the 

 horizon. 



In Dr. Button's Tracts there is given a problem for determining 

 the height ascended by a ball when projected vertically upwards with 

 a given velocity, and resisted by the air ; the force of gravity being 

 supposed to be constant, and allowance being made for the decrease in 

 the density of the air as the ball ascends. (Tract 37, prob. v.) In 

 the same tract there is also given (prob. xi.) an investigation of the 

 circumstances attending the motion of a body in air when projected 

 horizontally on a smooth surface so that the action of gravity may 

 produce no effect on the motion of the body, the resistance varying 

 as the square of the velocity. Also in Poisson's ' Traite de M^canique,' 

 the following remarkable circumstance is demonstrated : If a body be 

 projected, as in the last case, and if the resistance of the air vary as 

 the square root of the velocity ; the motion of the body will at first 

 diminish gradually till it becomes equal to zero ; and afterwards it will 

 go on increasing indefinitely. (' Tom.' i., no. 136, ed. 1833.) But for 

 the demonstrations of these problems our limits oblige us to refer the 

 reader to the works just mentioned. 



PROJECTION. The practical parts of this most important appli- 

 cation of geometry are noticed in the articles MAP; PERSPECTIVE; 

 GNOMONIC; GLOBULAR; ORTHOGONAL, ORTHOGRAPHIC; STEREOGRAPHIC; 

 ic. The present article is merely intended to point out the general 

 principle of all projections, and also to note the theoretical importance 

 of the subject. 



Imagine a surface of any kind, through every point of which passes 

 a curve the character of which depends upon that point, insomuch 

 that, given a point of the surface, the curve which passes through that 

 point is given in 'character and position. If any second surface be 

 taken, which is cut by all the curves emanating from the points of the 

 first, every point of the first surface has a point corresponding to it on 

 the second. Thus if the curve passing through A on the first surface 

 cut the second surface in o, the point A is said to be projected on the 

 second surface at a by means of the projecting curve A a. Similarly 

 any line on the first surface is projected into a line on the second, 

 which last contains the projections of all the points on the first ; and 

 the projections of the several boundaries of a figure on the first surface 



