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 be since the distance of the body from the centre diminishes, while the time reckoned from the moment of departure increases, we shall have d.x gx = - . This equation will be found to be verified by assuming x = a cos. t + b sin. t which being dif dx g ferentiated once, gives = V; in the equation for x, making x=r' (any given distance from the dx centre) when t= 0, we have a=r'; and in the equation for making dx Ti (the velocity) =0 when t= 0, we have Consequently x=r' cos, ; whence x is found when t is given : but when x=0, we have t 9 = (where w represents the half circumference of a r circle whose radius is equal to unity) whatever be the value of r. \, /r Therefore t= 2 i - - will express the time of falling from the surface g 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 vacua 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. (' Prineip.; 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 bo the space descended iu any time t : also if g be the force 1 1 " r=gof gravity at the earth's surface, ve shall have : :, ; (p 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 d=x 9",* therefore ate = (p- In order to integrate this equation, multiply both sides of it by 2dx ; 29r2 and then the first integral will be = + coast. The constant dt= px dx may be found on considering that (the velocity) =0 when t= 0, 1 1 when also x=0 ; therefore cone. = - p ' p-x p and -- =2gr= - ) ' x whence d7t-x = { 2p . p - x This equation may be put in the (p x) dx form = dt; and by the rules of integration we have Vpx P 2r= t /V = 1 p + - p arc cos. = 2x : there is no constant 2 to be added, because x=0 when t= 0. Prom this equation t may easily be found when x is given : likewise from the equation for dx 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 ohalf 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 be projected vertically in vacuo from the surface of the earth, and be subject to a variably accelerative force of attraction downwards. Let r 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 Iet h be the height due to the initial velocity, supposing the latter to have been acquired by a body falling in vaeuo with a uniformly accelerative force ; theu 2gA will express the square of that velocity. By the law 1 1 914 of attraction we have g : : 4. : + xr ; 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 --if In order to integrate this equation, multiply both sides by 2dx ; then we get wi = - r+x + coast.: and to find the constant, it must dx." be observed that x=0 when =20.; therefore mat. = 2g(h +r) 2g{hr + (A.+ r)x} and Tip. =2g (h+r)- or = -F x . Taking the square r roots and transposing, we have v{itr + (A+ r)x} dx = V2g . dt, and this equation may be put in the form (r+x)dx V {Ars +(2h+r)rx dt or (h being small compared with r) rejecting A when added to r, the equation becomes (r+x)dx V ihri = V2V dt.
Now, multiplying the numerator and denominator of the first member by 2r, and putting the whole in the form (r= +2rx)dx r dx 2r Vihr= + ex+ r.c1 2 N./{ + r=x + = . dt, the rules of integration give 1ra V{ltr= +r=x + re} + hyp. log.{ + A,/{ + r=x + re} + coast. t.
The constant may be determined by considering that x = 0 when t= 0; and thus t may be found when x is given.
What 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 a is the inclination of the plane to the horizon.
In Dr. Hutton'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 Traitf de 316canique,' the following remarkable circumstance is demonstrated :-If a body be projected, as in the last case, and if the resistance of the sir 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.