524 
FLUXIONS. 
CP 
CE j and confequently, as CB : CE :: x w (the ffdd 
CPyCE 
velocity) to 
C.B 2 
X v the angular velocity of the 
pci t E (revolving with the body.) The velocity at B 
BP 
in the direction CBE will be —— x v •' therefore, the ve- 
C B 
locity of E being to the velocity B, in the faid direc- 
PP y C' pT R "P 
tion, as -———- to *-, the fluxions of AE (z) and 
’ CB 2 CB’ 
CB ( x ) nuift confequently be in that ratio ; that is, 
CP x CE BP . . . CP XCE . be 
• -: ■—- :: z : a- ; whence z — —-:— '/i“- 
CB 2 CB CB x BP v 
vx 
abex 
abx 
b 2 
a ___________ _ 
*'*' x ^ %/x 2 v 2 — b 2 c 2 X\Jx 2 v 2 — b 2 c 2 x^J ' J 
Which equation in general, let the law of the centripetal 
force be what it will : but in the cafe above propofed, v 2 
being = p 2 -J—;— X «”+•*-——, and c 2 = />%"+*; it 
n- J-i 
n-\-i 
afpx 
becomes zz 
w. 
v 1 'I- 3 
P 2 -1 -— X x 2 — p 2 b 2 -— »+i 
«+1 . n -\-1 X « 
*= i 
P 2 X "+ 3 
X \ 
-P 2 b 2 + a . + i 
Lc. is reduced to a = 
ibx 
X J-b 2 +~ 
exprefled by ± ; according as the value of n Is pofi- 
tive or negative. 
Thus, if n — —2, and?«m, the body will fly entirely 
off in half a revolution : and, if n — —4, and tit 222 —1, 
it w ill fall to the center in half a revolution. 
Cor. 2. Moreover, though the fluent exprefiing 
the angle at the center cannot be exhibited in a general 
manner, yet there are certain cafes of the exponent (n) 
where its refpedtive values may be derived from each 
other. 
For let (as above) zz-^3 be put —m, and (to fhorten 
the operation) let CA ( a ) be taken as unity : then 
our equation will be transformed to z — 
bx ~ 
make y—x~„ 
whereof the 
truly defined by 
x V 1. 
2 X " 1 
=2 X x 2 — b 2 - 
in —2 ,p~ ’ m —2 .p 2 
and it will be farther transformed to z — ~ x 
b L 
y 
Put 
4 
+ 
whofe fluent is the meafure of the angular motion ; from 
which, when found, the orbit may be conftrudted : be- 
caufe, when AE, or the angle ACE, is given, as well as 
CB, the pofition of the point B is alfo given. But this 
value of z is indeed too complex to admit of a fluent in 
algebraic terms, or even by circular arcs and logarithms, 
except in certain particular cafes ; as when the exponent 
n is equal to 1, —2, —3, or —5; befides fome others 
wherein the values of p and n are related in a particular 
manner. 
Cor. i. If the given velocity at A be fuch that/> 2 -j- 
2 
ji -pi 
the value of is negative), our equation will become 
abpic 
..-p x y- 
-2 ./ “ 
and it will 
by 
— b 2 — 
2 y 
in —2. p 
become z =z — x 
m 
4 
Laftly, let 
— =, orp —^—— (which is always poflible when 
ry 2 r 
_ b 2 + 1 — -= 
r —2 .r 1 r — 
2 .p 
2 XX 
r—2. p 2 
or q 2 — 
(or 1 
r — 2.q 
2/> 2 
r — p 2 X r — 2 
by 
=) 
r—2.p 2 r — 2 -r* 
2 
and then we fhall have zzzz — x 
m 
X y 2 — i 2 ■ 
2 y 
Which exprefc 
r —2. q 2 
y \l x + = 
r — 
fion (excepting the general multiplicator - } being exafr- 
m J 
ly of the fame form with the firft above given, mu ft there¬ 
fore be t’ne fluxion of the angle at the center, when the 
index of the force is r —3 ; for the very fame reafons that 
the former appears to be the fluxion thereof when the 
index is in —3 (or n.) 
Hence, if the fluent of' 
by 
a/ I. 
fluent will be found equal to ± — multiplied by the 
difference of the two circular arcs, whofe fecants are 
i n 
. andf, to the radius unity. From this value of 
-2. q 1 
~ Xj V 2 —b 2 
2 y 
r—i-q. 
1 b 
ba 
the arch AE the pofition of the point B, in the orbit, is 
given. 
Bu,t if the angle of direction CA£ be a right one, the 
2d 
fluent will become barely =z ± — x arch, whofe fecant is 
711 
x h 
1- (becaufe then b=z.a, and the arch whofe fecant is “ 
k* ,7 
— o) which therefore when x 2 becomes infinite, will be 
X whole periphery AF, &c. 
Whence it is evident that the body mufl either fly intirely 
off, or fall to the center C, in a number of revolutions 
or the angle at the center, when the exponent is r —3 (or 
- _ -> ^ —-3) be denoted by &>, the value of z, 
m ft-J-3 
(the meafure of the faid angle, when the exponent is 
2 ZO 
m —3 (or n) will be truly defined by —. 
From which we collect that, if the indices of the force, 
in any two cafes, be reprefented by n and —-3, and 
n +3 
7 ,l ' 1 ~ 3 > 
the refpedtive diftances from the center by x and x 2 
then the angles t’nemfelves correfponding to thofe dif¬ 
tances will be every where in the conftant ratio of 2 to 
n- 1-3. Therefore, when the orbit can be conftrudfed ia 
the one cafe, it alfo may in the other, provided the above 
2 / ) ~ \ « 4-3 .p 2 
equation q 2 (=: ., —- 1 =-■ ■ ■ — — , for the re- 
lation of the celerities at A, does not become impoffible, 
as it will, fometimes, when n is a negative number. 
Cor. 3. If the body be iuppofed to move in a ver¬ 
tical 
