520 
FLUXIONS. 
that whereby a body might revolve in the periphery of a 
circle about the center C, at the diftance of CD, will be 
known ; for, if this laft velocity be denoted by w , the 
value of F will be rightly exprelfed by — : whence, by 
fubftitution, we have ± vv ='-, or ±: » 2 X—=a 2 X 
* v 
Thus, if in 
£ 
«ry , 
. (exprefting the fluxion of the 
y/ 2 2.*’' + 1 
time of defcent along AD) n be expounded by i, o, — i, 
and —3, iUcceffively, the fluxion itfelf will become equal 
to 
y/ %aXxx 
yf a 2 
y/ za — 2x 3/ ax—xx 
axx 
and — ■ —: 
y/a 2 
", and confequently w : v :: 
-: whence w 2 : v 2 : ± - 
* _ _ » 
l V I X • 
'V ± — : \ Where, as well as above, the fign of v 
v x 
mud be taken 4* or — according as the body is urged 
from, or towards the center C. 
Prop. LXXV .—Suppojing a body , let go from, a given 
point A zvitk a given celerity (e) along a right line CH, to be 
urged, cither way, in that line, by a force varying as any power 
(«) of the dijiance from a given point C ; to find, not only, the 
relation of the velocities , and/paces gone over, but ai/o the times 
of afeent and dc/ccnt, 
130.- The conftru&ion of the preceding problem being 
retained, F will here be expounded by x", and we (hall 
therefore have dc vv (zzzYx) xz.x"x ; and confequently, by 
Z)V X 11 * 
taking the fluent thereof, ±: —2= —,— ; but to correct 
0 ’ 2 n-\-i 
the fluent thus found, let x be taken = CA (which we 
will call a) then v being =c, the fluent in that circum- 
C ^ Q n " 4 “ ^ 
fiance will become ± — ——;—: therefore the fluent 
fpedtively : whence, if ARF be a quadrant of a circle 
whofe center is C, and ASC a ferni-circle whofe diameter 
is AC, and DSR be perpendicular to AC ; then it will 
appear, 
- x A r 
x. That, when n—i, and T =— ._ , p 
\J a 2 —x 2 
the velocity, (3/ a 2 —^ 2 ) at D will be 
reprefented by DR, and the fluent fought C 
, AR 
by 
AC ' 
2. That, when n—o, and T 
the velocity 
yj za—zx 
at D, and the time of defcent through AD, will each be 
defined by y/ 2AD. _ 
3. That, when n — — 2, and T — ULkUL., the velo- 
■f ax—xx 
and the time of 
f J ax — xx\ 
city ( 1 - will be ; 
\ x f {a J 
DS 
CDp/JAC 
n-\-i ’ 
defcent through AD, as yj £ACx AS + DS. 
V i 
duly corredted is ±; — qz 
2 
2X»+ 1 C/) 20"+ 1 
» 5 oo c 2 = 
- *— a n + 1 
4. And that, when n — —3, and T =■ 
n-\-i 
2. n 
whence v will come out 
DR 
•;/ a 2 — x 2 
, the 
-, and the time as ACxDR, 
zp. za ”-^ 1 dr zx nJ r 1 
n-\-1 
where the figns of v and a'”-M 
iniift be alike, when both quantities increafe, or decreafe, 
at the fame time ; that is, when the force, from C, is a 
repulfive one ; but, unlike, when one increafes while 
the other decreafes, or the force, tending to C, is an 
attractive one. In the former cafe we therefore have 
velocity will be as ^ ^CD ’ 
Hence the time of the whole defcent through the ra- 
AF - - 
dins AC, appears to be as f/^AC, i/|ACxAF, 
A 
or AC 2 . But the time of one whole revolution in the 
4 A F 
periphery ARF, &c. was found to be as- 
c 2 + 
2X ,, + 1 - 2 / 7 “+ 1 
n- f- 1 
j and, in the latter, v = 
in the four cafes above fpecified is 
, 4 i 1 
AC 2 
4 A F, 4 A F 
"AC 
which 
4AF 
4 *-O 
Hi 
The value of v being thus obtained, let the required 
iime of moving over the fpace AD be now denoted by 
T; then, fince T is univerfally =- 
/ 2X" 
x "+ 1 — za + 1 
=5, or T = 
we have T 2= 
x 
4 
za”'r 1 ■—2x”+ 1 
ac- 
n-\-i 'S ' 1 K+x 
cording to the two forefaid cafes refpedtively : from 
whence, by finding the fluent, the time itfelf will be 
known. Q^E. I. 
Cor. If the body proceeds from reft at A, c will be= 
- ■ n L 
©, and we fhall have T = 1 + h 2 X x ^ or T 22: 
V 2X"t 1 — za’‘ J r 1 
+jf 2 X 4 - 
3/ 2l 1 -2X"+ 1 
Scholium. —Although the fluents of the expreflions 
given above cannot be exhibited, in a general manner, 
neither in finite terms, nor by means of circular arcs and 
logarithms; yet, in fome of the mod ufeful cafes that 
occur in nature, they may be obtained with great facility. 
VAC’ 
X y/ AC, and 4AFX AC : therefore, if the time of moving 
over the quadrant AF be denoted by it follows that 
the time of defcent through the radius AC, will be truly 
ACt/T — AC 
defined by Q_, Q_X QjXVb or Q-X aF 
according to the aforefaid cafes rcfpeflively. 
Lemma.— The areas which a revolving body deferibes, 
by rays drawn to the center of force, are proportional to 
the times of their defeription. 
151. For, let a body, in any given e D 
time, deferibe the right-line AB, C T : -T^ 
with an uninterrupted uniform mo- g'( 
tion ; but upon its arrival at B let 
it be impelled towards the center A S 
S, fo that inflead of proceeding along ABC, it may, after 
the impulfe, deferibe the right-line Be. 
Becaufe the force, afting in the line SB, can neither 
add to, nor take from, the celerity which the body lias in 
a direction perpendicular to that line, the diftance of the 
body from the faid line, at the end of a given time, will 
therefore be the very fame as if no force had acted ; and 
confequently the area B^S equal to the area BCS, which 
would have been deferibed in the fame time, had the 
body proceeded uniformly along BC ; becaufe triangles, 
having the fame bafe and altitude, are equal. 
. Therefore, 
