FLUXIONS. 
303 
pofed to be diminifhed ad infinitum, that-their firft ratio 
may be always obtained. 
4. In the refolution of any problem, the nature and 
conditions of it are to be elofeiy examined and ftridtly 
pttrfued according to all the known methods ot algebraic 
reafoning, by attentively confidering the relations of the 
quantities, and their mutual proportions and dependance 
on one another ; and forming your procefs according to 
thefe their properties, by duly comparing together the 
quantities, their moments or increments, their fluxions 
or fecond fluxions, &c. as the cafe requires; till you get 
a competent number of equations or general proportions. 
And then you muff proceed to expunge fuch quantities 
as are fuperfluous, till at laft you get a fluxionary equa¬ 
tion or proportion with the quantities required. Then, 
if there be occafion, 
5. Find the fluent of the faid equation or general pro¬ 
portion : and then you have a complete equation or ge¬ 
neral proportion containing the quantities fought. 
6. But to obtain an equation of the indetermined 
quantities, by having the corredt general proportion be¬ 
fore found, or by having only the fluxionary proportion ; 
you muft aflign to each indetermined quantity in the faid 
proportion, (or in the fluxion thereof,) fuch a deter¬ 
mined value as it is known to have in any particular 
cafe ; and from thence you muft draw an analogy from 
jhe fluxion alone (of the general proportion), or from 
the corredt fluent alone, (or fometimes from both toge¬ 
ther.) From whence there will be had an equation be¬ 
tween the quantities required ; or at leaft between their 
fluxions, whofe'fluent muft then be found and corredted. 
And note, thefe determined values (of the quantities) 
may be either exprefled in numbers or fymbols, as any 
one (hall think proper. Sometimes it may be fufficient 
to aflume a given quantity, by which multiplying one 
fide of the proportion it will be turned into an equation ; 
and this given quantity may afterwards be determined 
according to the nature of the queftion.—Thefe are the 
general rules, but after all, many things muft be left to 
the fagacity and invention of the artift. 
Cor. —Hence every problem belongs to fluxions, in 
which the increments, or the proportions of the incre¬ 
ments or moments of the feveral variable quantities con¬ 
tained therein, can in all cafes be computed and exprefled 
by equations. 
Ex. 1. To find the velocity wherewith the ordinate 
BM of a circle increafes in every point, whilft it moves 
uniformly along the diameter AD. 
Let AD—2r, AB=r, BM=ry; and by the na- 
zrx—xxzzyy ; this equation put into 
fi uxions gives lrx — zxxzzi,yy, 
f' _ x 
or-- xz=.y y a general equa- 
y 
tion for the increafe of y in 
all points. Therefore in A 
T - X 
I) whereby is o, and x is o, ——— 
TX 
3 czzy becomes — zzy, therefore y is infinite. If CB = 
BM, or r — xzzy, then 3 c—y, and x and y increafe 
equally. But in C where r — x—o, then y —o, therefore 
y does not increafe at all. In b where Cbzzbm, then 
jzz — 3 c, therefore y decreafes as fall as x increafes. Laftly, 
——TX 
in D where y—0, and xzzir, yzz -=2— infinity, and 
there the ordinate decreafes infinitely : and in all the in¬ 
termediate points it has all the intermediate degrees of 
increafe or decreafe. 
Ex. 2. To find the fpace a defeending body will de- 
feribe in any time by the uniform force of gravity. 
123. Let zzz fpace, x— time, V— velocity, acquired in 
. c~ III 
$bat time. By the principles of mechanics z ot vx « xx 3 
122. 
Jure of the circle 
and therefore z oc vx at xx ; that is, the fluxion of the 
fpace is every whereas the velocity into the fluxion of 
the time ; that is, (becaufe the velocity is as the time) 
as the time into the fluxion of the time. 
Now if only the ratio of the fpace and time berequired, 
x 2 
it will be fufficient to take the fluent; and then see—, or 
2 
zeex 2 , that is, the fpace is always as the fquare of the 
time. But if the abfolute quantity of the fpace is 
fought, we muft reduce the general proportion or its 
fluxion to an analogy from fome particular known cafe. 
Thus it is known by experiment that in the time t or 1 fe¬ 
cond, a body would acquire fucli a velocity as to move 
through s or 32-tTeet uniformly in that time, or to have 
defeended through y or 16-j^ feet in that time. Whence 
sx , 
t : s :: x : — — fluxion of the fpace 2 when xzzt\ where- 
sx 
fore from the general analogy (ia«) we have y : tx 
i, : xx, and zz 
. S —, and finding the fluent, zz r. —— 
it ° 2 tt* 
which needs no corredtion (becaufe when zzzo, xzz o). 
sx x 2 
tx :: z : ■—. 
Or thus from the fluxion and fluent, 
whence z— - 
2 tt 
Or laftly, thus: Since %szzz, when t—x, 
therefore from the general analogy {zatxx) it will be 
sxx 
bs : tt :: z : xx: whence zzz -, and thus the fame 
2 2 tt 
equation is obtained any of thefe ways. 
Ex. 3. If a body is projected upwards with a given ve¬ 
locity a, to find how far it will afeend in any time x. 
124. Let zzr;fpace, v— its velocity; then by Mechanics 
zCtvx. Now fince the firft velocity is given, therefore the 
fpace s, which would be uniformly delcribed in the time t, 
_ i 
of its whole afeent, will be given; wherefore t : s :: * : 
/ 
S X / 
moment of the fpace z at the firft inftant, that is, 
when a — v. Therefore from the general proportion 
(2 Cf.vx,) ax :: z 
vx : whence z—~ 
ta 
but the velocity the body lofes is as the time, therefore 
clx , ax 
t ; a :: x : — zr^velocity loft, whence vzza -. there- 
t t 
sx sxx 
fore zzz — — -— 
t tt 
and the fluent zzz —— 
t 2 tt 
which 
needs no correction. Hence if x be greater than 2 1 , the 
body will have defeended again below the point it was 
projedted from. 
Otherwife thus.—Let zzz fpace, vzz velocity, szz velo¬ 
city generated by gravity in the time t ; then by Mecha- 
II 
nics zQCvx. Now fince the firft velocity is given ; there¬ 
fore the fpace d , which would thereby be uniformly de- 
feribed in a given time t 9 will be given $ wherefore 
/ 
# dx / 
t : d :: x : — =2 moment of the fpace z at the firft in. 
t _ 
ftant, that is when azzv. Therefore from the general 
' • dx 4 4 1 . 
proportion (z&vx) — ; ax :: z : vx :: z : vx, whence 
»=: —j but the velocity the body lofes is as the time ; 
ta 
therefore 
