FLUXIONS. 
500 
therefore t : s x 
s x „ dx 
—, therefore - 
t ’ t 
S X « 
— == velocity loft; whence v—a — 
cbxx , , „ dx dsxx 
-: and the fluent z— -: 
att t tatt 
butt : d :: i : «=-: therefore zxzax ——". Hence 
t 2 1 
if zta be greater than sx, the body will have defeended 
again below the point it was projected from. 
Ex. 4. To find the time wherein a given cylinder 
found the fame way, only by fubftituting the value o 
EF or y had from the nature of the curve, into the 
—ty 2 x 
equation i— -—, and then finding the fluent, where 
d <J hx 
d— fiquare of the diameter of any cylinder, and h— 
height, t— time of the cylinders running out with that 
firft velocity. 
Ex. 6. Let ACE be (the fedlion of) a wall fupport- 
ing a fluid behind it, and joining to the perpendicular 
full of water will empty itfelf bv a hole at the bottom. AC; *° find the curve ADE teiminating the other 
— -— . ' .. ^ . . * flf f.v'ill f/% ♦hnt ifp rontYI n tt'i n if r> A n lr o f ,1. U .. .-n 
125* Let A C=zh, CE=x, AE: 
x, t— time of run- 
fung out with the firft veloaity, 2— time fought. 
Now the moment of the quantity 
out 
OC2X velocity; but 
is as 4/CE, and the 
running 
F the velocity 
moment of the quantity is as the 
moment of AE, or —. 
11 1 
—xC(.zV x > therefore zO. 
whence 
—-X 
-j/w ’ 
but 
h : t :: —x 
firft inftant. 
1 
-t x 
1 — 
kCX-t 
V 
=)■ 
-tx 
~x 
V' k 
—# 
■y/ x 
■—X 
V x y 
whence z= 
m—tX 
-——, and the fluent is z— 
y hx \fh 
G, = o, and x — h, therefore 
2 tx^ 
—; but in the point A, 
the fluent corrected 
is Z — it 
2 tx* 
and when x=o, then the whole 
—dx -j —— 2 
-X b-\-x . 
PP 
b + x 
—dx 
PpV* 
—tx 
pp y hx 
then 
— tx 
~T~ 
—dx 
y h 
Therefore z C£ —— X 
PP V* 
I 
1 — dx —— 2 
0—1“ X ! S 2 ! 
PPV* 
y^b-\-x , from the general analogy; therefore i=z 
:XM“* • And the fluent is 2= 
— t 
'pPV /l 
X 
2 War 2 
3 ^ • 
; and when corrected, the whole time z—ty. 
^ n( j jf t ] ie f ru ft, um were inverted. 
PP 
the time would be found to be 
moment of the time at the 
Therefore (from the genera? proportion 
time z—it. 
Ex. 5. To find the time wherein a given fruflrum of 
a cone, full of water, will empty itfelf by a hole in the 
bottom. 
126. Let the cone be completed, and put VGzr^the 
altitude; TG —/4 the height of the fruftrum, TS^rx, circle 
CD=if, VT z=.b, t— time of running out 
-Gr-I) ( 0 f a cylinder whofe bafe is d and height 
h) with the firlt or greateft velocity, 
z — time fought. Proceeding here as in 
/ 
the laft example, you will find 21/x CX mo. 
/ 
ment of the quantity CC —x X circle EF a 
. I 
—dx 
... ipp—fpk + Zhh 
lX bb 
Where d=z circle AB, x=GS. 
Scholium.—I f EF or y be the double ordinate in any 
curve CA ; the time of running out might have been 
Vol. VII. No. 445. 
fide of the wall, fo that its ftrenglh may be every where 
as the prelfure it fuftains. 
127. Let AC=/i, AB—re, BD=±y. The effeft which 
any number of particles of the fluid prefling at B have 
to break the wall at C, is as CB X 
number of particles x their force, that 
is, a h — x XxX x, (becaufe the num¬ 
ber is as x, and the prelfure or force as 
x). And the fum of all the forces 
adting on AB to break it at C is as the 
fum of all the h —.vX« that is as the fluent of hxx — x 2 jv, 
hx 2 x 3 
and therefore as-, and when x=.h, the whole 
23 
prelfure on AC to break it at C will be £k 3 ; therefore 
the effects of the prelfure at B and C will be as AB 3 
and AC 3 . But the ftrength of the wall in B and C is 
fuppofed to be as thefe forces, and by Mechanics it i^£ 
known to be as BD 2 and CE 2 ; therefore AB 3 : AC 3 :: ' 
BD 3 : CE 2 , that is jy 2 C<x 3 J and the curve ADE is a 
femicubical parabola whofe vertex is A, and therefore 
convex towards AC. 
Ex. 7. Suppofe a wind to blow againff the perpen. 
dicular fide AC of the wall ACE; to find the curve 
ADE bounding the other fide, fo as the ftrength of it be 
every where as the force it fuftains. 
128. Let AB=x, BD =zy, AC —k. The force of any 
particles at B to break the wall at C is as CB x number 
of particles 0./1 —*x*or CS.k —xX-v; and therefore the 
whole force of all the particles onAB to break the wall at 
2 
C is GC the fluent of hx—~xxOihx -—; therefore the force 
2 
to break it at C by all the particles on AC is \hh or as 
hh, and this mull he as the ftrength of the wall or as 
CE 2 ; confequently x 2 C£ y 2 and xCC y, therefore ADE is 
a right line. 
Ex. 8. Let ABC be a heavy body, BC a fpring fixed 
to the block D : Let AB be clofe thruft up to C, that 
the fpring may be clofe bent, and fixed thus to the (lock 
D by a pin. To find with what velocity the body will 
be projected by the force of the fpring when the pin is 
fuddenly plucked out. 
129. Let B C—b the length of the fpring in its natural 
pofition, CN—x, <v— velocity of the point B when it ar¬ 
rives at N, w— weight of the 
body AB, t— time of deferibing 
CN. By Mechanics or the Laws 
of Motion va. 
DUtiJ 
the nature of a fpring) -—- t 
zo 
g 
likewife by the Laws of Motion t<j. ‘ ; therefore 
® ZO f 
X - or et'CC- 
V 
bx—xx 
, and finding the fluent, w 2 Ct- 
4 G 
2 bx —.fir 
eO 
and 
