FLUXIONS. 
tidal direction AH: then* putting the velocity 
575 
/ "-M 
p~ F —j-— x 
n- j-1 k + x 
l 
2= o, we get x (CH) = 
tually be the cafe when the value of p 2 
n -\-1 
is potx- 
but not otherwife, the 
tive, or p 2 greater than 
fquare root of a negative quantity being impoffible. 
Thus, if n — — z, or the force be inverfely as the 
fquare of the diftance, and p 2 , at the fame time, greater 
than 2 (==) the body will not only continue to af- 
cend in infinitum , but have a velocity always greater than 
that defined by p 2 —2, which is its limit. 
Cor, 6. Hence the leaft celerity fufficient tocaufe the 
body to afcend for ever in a right line is given. For, 
T . . I 
n -\-1 
ap- 
he 
putting */ /i 2 4- 1 - X a =0, we ha ve p — J - 
' Tlfi-l ^ —72—x 
Therefore the leaft celerity by which the body might af¬ 
cend for ever, is to that whereby it may revolve in a 
circle AEF, as J -to unity. From which it 
pears that, if the force be inverfely as any power of th 
diftance greater than the third, a lei's velocity will caufe 
a body to afcend ad infinitum than would retain it in a 
circle. 
Scholium. —From 'he ratio of the velocity 
u 
P~ +—;-X a’T 1 
1 
VoL. Vli, No. 446, 
2 X n ^ * \ 
---] wherewith the body ar- 
• n + ij 
) 
TT- , A + y. a — the height to which the body 
hi ' 2 X « 4 -i + M 
_ __ 1 
will afcend: hence hp 2 yn 4- 1+1^ ' ,+ ‘ X <z —a (=AH) 
is the diflance through which it muff freely delcend to 
acquire the given celerity at A : this diftance, in cafe of 
an uniform force, when n—o, will becom o — \p 2 a : and, 
when the force is inverfely as the fquare of the diftance, 
it will then be 2= —-. 
, 2 ~P 2 
But, when p— 1, or the velocity at A is juft fuffi¬ 
cient to retain a body in the circle AEF, AH becomes 22: 
LjXl M-i X a — a: which in the two cafes aforefaid will 
2 I 
be equal to and a refpedtively ; but, infinite, when 
n is =: —3- 
Cor. 4. When the value of n-\- 1 is pofitive, the ve- 
lo ritv at the cente r, where x—o , will be barely 222 
fi 2 +—-— X «"+*; but if the value of rc-J~i be nega- 
nfi- 1 
tive, the velocity at the center will be infinite ; becaufe, 
then o"+* is infinite. 
Cor. 5. Moreover, when n + i is negative and x infi¬ 
nite, the velocity alfo becomes =2 J /> 2 + .. 2 .. x<z’ ,+ 1 5 
» n 4-1 
becaufe then *”+‘2=0. 
Hence, if the centripetal force be inverfely as fome 
power ot the diftance greater than the firft, the body may 
afcend, ad infinitum , and have a velocity always greater 
1 ■ ■■■ — . 1 
than 'VP 2 -}-—v a +i > which is to, pa 2 , the given 
72 + I 
velocity, at A, as 'J p 2 -i- —-— top. And this will ac- 
, 1 
rives at any diftance * from the center, to that . 
V * 
which it ought to have to revolve in a circle at the fame 
diftance, it will not be difficult to determine in what cafes 
the body will be forced to the center, and in what others 
it wdll continue to fly it ad infinitum. 
For, firft, if the angle CA b be acute, or the body from 
A begins to delcend, it will continue to do fo till it ac¬ 
tually arrives at the center, if the former velocity, during 
the defeent, be not fomewhere greater »han the latter, or 
the quotient + _L_ x ^7 — ~' greater than 
unity ; becaufe, if it ever begins to' afcend, it muft have 
an apfe , as D (where a right line drawn from the center 
cuts the orbit at right-angles), and there the celerity muff 
evidently be greater than that fufficient to caufe the body 
to revolve in a circle. 
Secondly, but if the quantity J r - X +1 n+l > 
in the accefs of the body towards the center, increafes fo 
as to become greater than unity, or be every where fo ; 
then the velocity at all inferior diftanees being more than 
fufficient to retain a body in a circle at any fuch diftance, 
the projectile cannot be forced to the center. 
After the fame manner, if the angle C AZ> be obtufe, or 
the body from A begins to afcend, it will continue to do 
fo for ever, when the forefaid quantity is always greater 
titan unity, or, which is the fame, when the body, in its 
recefs from the center, has in every place through which 
it paffeth, a velocity greater than fufficient to retain it in 
a circle at that diftance. 
It therefore now remains to find in what laws of the 
centripetal force thefe different' cafes obtain : and, firft, 
it is eafy to perceive that when the value of n + i is poli- 
p 2 +. 
a" + 1 
X-TjT- 
tive, that of \ p 2 4- 
<z’ l + 1 
X 
n + i 
" + * 
n-fii 
will, by in- 
creafing x, become equal to nothing. Therefore the body 
cannot afcend for ever in this cafe : neither can it defeend 
to the center (except in a right-line) becaufe the forefaid 
quantity, by diminifhing x, becomes greater than unity 
(or any other aftignable magnitude.) 
But, if the value of n be betw ixt —1 and —3, the faid 
general expreflion, taking x infinite, will alfo become in- 
2 
finite, provided the value of p 2 4- - be pofitive (or p 2 
2 
greater than-. Therefore the body, in this cafe, 
may afcend ad infinitum t but cannot poftibly fall to the 
center (except in a right-line) fince, \ -—, the value 
n 4-1 
of the general exprefficn, when x=zo, is ^greater than 
unity. 
Laftly, if n be exprefled by any negative number great¬ 
er than —3, or the law of the force be inverfely as any 
power of the diftance. greater than the third, the two ex¬ 
treme values of «Jp 2 + 
,41 
n-\-i 
X 
nfi-i 
will,> 7 /, be 
denoted as in the preceding cafe; but here the latter of 
them, is lefs than unity. Therefore the body 
muft, in this cafe, either afcend for ever, or be forced to 
the center; except in one particular circumftance, here¬ 
after to be taken notice of. 
Now', from thefe oblervations we gather, 
1. That, when the centripetal force is as any power o.f 
the diftance direftly, or lefs than the firft power thereof 
6 S inverfely* 
