512 
FLUXION S. 
ABx^, draw the tangent BS, and BR=BA, perpendi¬ 
cular to the horizon, and RS parallel to it. The line 
BA is fuftainea by three forces, for its gravity a6ts in di¬ 
rection BR, it is drawn at A in direction AZby the force a, 
and it is fuftained' in B by the tenfion of the line in direc¬ 
tion SB ; and thefe three forces being as BR, RS, and BS, 
and BR=z by conftrudlion, therefore the force a=RS : 
whence by fimilar triangles BR ( z ) : RS (a) bo ( x ) : 
Bo (y) :: x : y, and ax=zy, which is one property of the 
curve. 
3. Take Br—Bb the increment of the curve, draw rn 
parallel and Bn perpendicular to BS, then is rcBrerBo, 
and rnzzbo. Since BR is the perpendicular force or 
/ 
♦eight of the line at B, Br or z is the increment thereof, 
and therefore x is the increment of the tenfion of the 
line in B, and x is the fluxion of the tenfion, and there¬ 
fore the fluent x— tenfion or force adting in diredtion nr ; 
but in A where xz=o, this tenfion z=«, therefore by cor¬ 
relation, the whole tenfion drawing in direction of the 
Curve is a-\-x\ and this is the force BS, as was (hewn be¬ 
fore: therefore again by fimilar triangles a-\-x (BS) : z 
CBR) :: z' : x :: a : x, whence ax-\-xx=zzz, and the 
fluent 2ax-\-xxzz.zx, which is another property of the 
curve. 
4. If the point B be fo taken that the angle RBS or 
SBC be half a right angle, then will AB or z be =<2; 
f ax \ 
for tjien x—y, and f -y— la. 
c . . ax 
3. Since y —— 2= 
x 
3/ 2 ax-^-xx 1 / aa-\-zz 
a-\-x ’ 
(therefore y— 2.30258 ay, log aJ r x -\ m V 2 *- v + a-y— 2.30258 
0 X log 
z-J-p/ aa-\-zz 
; whence the curve may be eafily 
»CC- 
XX T X X • ltp ** ' ■'»'- 
and --— or vy OC r — x x \ and finding 
. „ vv 
the fluent — CC 
2 
in A, *2=0, therefore the fluent 
corrected is twCC 
But the fluxion of the axis isas ® 
the velocity of a defcending body, and the fluxion of the 
ordinate is as the fluxion of the time, or as a given quan. 
tity b\ therefore x : y :: (2=) 3/ 
bx 
1 —r— a; 
+ * 
whence y—- 
72 4-1 
bx 
V 
[ . — + 7 
#•”+ 1 r—x 
or y- 
V 
rr t -"+* 
r” + 1 *— r—x 
72-|-i n-\-i 
and the fluent will give the nature of the curve. 
bx 
Cafe 1. Let ?2=:i, thenj2=— -- : defcribe the 
3/ zax—xx 
quadrant AEF ; then by 
the nature of the circle, 
TX 
arch AE222 fluent of- , D 
\E 
conftrudled. 
Ex. 13. To find the nature of the curve BM in 
which a body moving (after its fall through AB), it 
fhall defcend equal fpaces in equal times. 
135. Let ABzza, BPz=.v, PM== y, now fince the velo¬ 
cities of bodies are as the fpaces 
A defcribed in equal times, and the 
fquares of the velocities are as 
the heights fallen from ; therefore 
a : a-\-x :: (fquare of the velocity 
in the axis at P : to fquare of the 
velocity in the curve at M :: Pp 2 : 
Mt 72 j :: x 2 : x 2 -\-y 2 ::) x 2 : x 2 -\-j> 2 , 
and by divifion a : x :: x 2 : y 2 ; 
1 1 
therefore xx 2 —ay 2 , or x 2 x—a 2 y, and 
3 
finding the fluent %x 2 —a‘-yfrov- ay 2 
2=-j.v 3 . Therefore the curve is a 
femi-cubical parabola convex towards BP. 
Ex. 14. If a body be projefted from any point A 
parallel to the horizontal plane BG| and be urged 
towards that plane with a force which re as any power of 
■its height above the plane ; to find the nature of the 
curve it will defcribe. \ 
136. Let ABirzr, AD—x, DP—_y, v— velocity acquired 
in falling through AD, *=2time of falling, y— force in 
D or P, which fuppofe to be as BT>. Now from the 
» 
, , I I IX -I 4^;. 
daws of motion v&ft, but and /our—x , ther cftm 
3/ 2 rx—xx 
b 
therefore take y or DP=-X 
r 
arch AE, and P will be in the R 
curve required. 
lx 
Cafe 2. Let 22=20, then y——-~, and yzzib\/x t or 
V x 
ybbx—yy ; therefore the curve will be a parabola, as is 
well known. 
Cafe 3. Suppofe 722=—1. Here we mull have re- 
courfe to the original procefs 
II 
-, defcribe the hyperbola HE 
to the affymptotes AB, BC; D 
then the area ADEH— fluent of 
x - 
-; therefore aCC-i/ADEH, 
r—x v 7 
bx 
and_y2=- 
3/ area ADEH 
ture of the curve AP. 
for the na- 
E \P 
B 
C 
Cafe 4, Let 722= —2, then yxzbxy/ - 
be a cycloid, AG—v, DE= u, AGB 
being the generating circle ; then * 
22 2= v -f- 3 / rx — xx, and u =2 v 4- 
r —zx 
- x 2= (becaufe v = 
23/ rx—xx 
rx ... r—x 
Let AE 
Z,)xy/ 
2 y rx — xx * 
But y—br ' 2 
yx\/ -, whence yzzbu^r, therefore fake DP ■xz.h^/ry 
DE, and P will be in the curve. 
~ r . bryrx — xx , 
Cafe 5. Let 72222—3 j then y ~— , - . whence 
3 / 2 rx—xx 
yzzbr 
