Sfifi 
FLUXIONS. 
5 XQjC C-* 2 ) and confequently v : so _ 
J**i 
:: J— : ,J-iL (becaufe R 2= — ^ 
* u U / 
T!ie far 
!. Prop. ] 
The fame proportion may alfo be derived from Corol. 
a. Prop. Ixxiv. For it is there proved that v : zo :: 
and it appears from above, that — 
C =•: 
and 
Ibss 
— X 
\ a bss 
whereof the fluxion being 
2 U 
a 3 
(inftead of -x & 2 ) would have been 5 and fo 
u 2a i 
= ^ x 75 
ps' 
the very fame as before. 
been 
» u 
-—: whence the whole is manifeft. 
v u 
If OL be made perpendicular to QC, QL will be 
CPx QO \ mR QL «R 
{—— - : J =-, and -= ——; and therefore v: 
* CQ_ J s CQ_ j 2 
t I . ; 
zu :: QL 2 : CQ^ : which is another proportion of the 
propofed celerities. 
Cor. 4. Lafily, the law' of centripetal force being 
given, the nature of the trajeftory AQjmay from hence 
be found; for fince the force (F) is univerfally defined 
u ' -— 1 
by —r it is evident that -will be 2= the fluent of 
u 3 s 2 u- - 
Fa ; which, when F is given in terms of s, will become 
known ; and then, the relation between u and s being 
given, the curve itfelf is known. 
Ex. 1. Let (he given curve AQH be the logarithmic 
fpiral, and C the center thereof: then u (CP) being in 
,. r bt u bs a 3 \ 
tins cafe = —’ we have- (=—Xttt I 
a 7* ’ 1 
u 3 s as * 
a \ 
— j =umty. 
bss s 
b 2 s 3 ’ “““ J— ( 
A N su 
Hence, it appears that the force is inverfely 
as the cube of the difiance ; and the velocity, every 
where, equal to that whereby the body might revolve in 
a circle at the fame diflance. 
Ex. 2. Let it be required to find the law of the cen- 
Jripetal force, whereby a body, tending to the focus C, is 
made to revolve in the periphery of an ellipfis AQDB. 
From the other focus F draw FK parallel to CP meet¬ 
ing the tangent PQ_(at right-angles) in K; join F, Q_; 
O^VK. putting the tranfverfe axis ABz^a, the 
femi-conjugate OD =2 £ b, and the para- 
AC OO meter : then, CQand CP being 
denoted as above, we have FQ^(—AB—CQ^) 2= a—s 
whence, by reafon of the fimilar triangles CQP and FQK, 
it will be s : u :: a—s : FK22-—But FK x CP 
is== OD 2 (by the nature of the curve. Hence we get 
.2— sxu 2 1 4* 4 
-2= and confequently — 22:-—■— 
3 H ’ u 2 b 2 s b 2 ‘, 
But, had it been a parabola, the equation would have 
Q l — q 2^2 ^2 \ 
- xu 2 z=.\b 2 , or —• (22:— ) 2=^ p ; and the 
s s /\.a J 
force, ftill, as . But, the meafure of the velocity 
ps 2 
^^ us — -^j 20 in this cafe becoming barely 222 
4/2, it follows that the velocity in a parabola is to that 
whereby the body might defcribe a circle at the fame 
diflance from the center, in the conftant ratio of 4/2 
to unity. 
Ex. 3. Let it be required to find the law of the cen¬ 
tripetal force, by which a body, tending to any given point 
C, in the axis, is made to defcribe a conic fection AQH. 
154. Put the femi-tranfverfe axis (OA) 2= a, the femi- 
conjugate —b, and the given 
H diflance of the point C from 
the vertex Azitc. - put alfo the 
abfcifla AE —x, the ordinate 
EQ=y, and CQz=s f as before.) 
The area of the triangle 
ECQbeing (22iECxEQ.) =2 —-—, its fluxion is there- 
N 2 
cy — xy—yx 
fore =z - - -*; which added toyx, the fluxion of the 
cy-\-yx—xy 
area AEQ, gives ^—---- for the fluxion of the whole 
2 
area ACQ^defcribed about the center of force. Whence 
{by Art. 152.) the required centripetal force at Qjvill be 
=p. Which expreflion is general, let the 
, we obtain 
u 
U 3 J 
f “ b 2 X 
-L=~,and 
s 2 ps 2 
jus 
1 2 x <2 s __ / 
J •—A 
su 
la ~ V 
Hence, it appears that the centripetal force is, in this 
cafe, as the fquare of the diflance inverfely ; and the ve¬ 
locity at Q^is to that whereby the body might defcribe a 
circle at the diflance CQ^, every where, in the ratio of 
FQ^ to AO 2 "* 
If the curve had been an hyperbola ; then - —7 X « 2 
~ sx y 
<yf-yx—xy 
curve be ot what kind it will. But in the cafe above, y 
being — --f 2 axzhx 2 , we have y 
bx'Xadz: 
a a\/ 2axdbx 2 
' .... . bxXoa-i-ax±cx 
__and cy-fyx—xy =z - ; and there-. 
2axzh^x 2 i ^ f 2ax^jzx 2 
fore, by fubftituting thefe values, we get 
—sxy 
cj-\-yx—fp 3 
<2 4 
3. Which, becaufe -p is conftant, will 
From whence it follows, 
AO- 
b 2 X Trfl-tzxTEcx) 
s 
alfo be as — ■-■ =73. 
ca-\-axphcx\ 
1. If cbe = +«, or the center of force be in the cen¬ 
ter of the fedlion, the force itfelf will be barely as (qrr) 
the diflance. 
2. If it be in the focus, then ac-fax±cx becoming 22: 
CQ Xa, the force will be inverfely as the fquare of the 
diflance. 
3. If the given point be in the vertex A, the force will 
S J 2 \ 
be as— : which therefore in the circle (where *22- 1 
X 3 20. 1 
will be as -L, or the fifth power of the diflance recipro- 
•s 5 
cally. 
4. Laflly, if the point C be at an indefinite diflance 
from the vertex, or the force be fuppofed to aft in the 
diredfion of lines parallel to the axis AO ; then the force 
will be as the cube of OE inverfely. 
Prof. 
