334

MISCELLANEOUS

Tract 3s.

^rV a g X (AB + bp). And when x = r, this becomes

t = kJ— X for the whole time of falling to the sur-

face at s; which is evidently infinite when a or AC is infi-nite, though the velocity is then only the finite quantityv/4 gr.

When the height above the earth’s surface is given g;because r is then nearly = a , and ad nearly = ds, the timet for the distance g will be nearly --------

V-~r x 2DS = x ^ 4 S r — as 5t ou g hli t0 be -

If a body, at the distance of the moon at A, fall to theearth’s surface at s. Then r — 3965 miles, a = 60r, andt = 416S06" = 4 da. 19 h. 46' 46", which is the time offalling from the moon to tiie earth.

When the attracting body is considered as a point c ; thewhole time of descending to c will be - -- -- --

l , a -7&54a .a lO/i , ‘7854 , fl 3

zrv~ x abdc =—-v— = = - v —'•

Sr g r g blr r g

Hence, the times employed by bodies, in falling fromquiescence to the centre of attraction, are as the squareroots of the cubes of the heights from which they respect-ively fall.

PROBLEM XIII.

The force of attraction below the earth's surface being di-rectly as the distance from the centre; it is proposed to deter-mine the circumstances of velocity, time, and space fallen by aheavy body from the surface, through a perforation madestraight to the centre of the earth; abstracting from the effectof the earth's rotation , and supposing it to be a homogeneoussphere of 3965 miles radius.

Put r — ac the radius of the earth,x — cp the dist. from the centre,v = the velocity at p,t — the time there,g = 16 L;, half the force at A,f = the force at P.