756 
be y 
FLU 
FLU 
a + a ) * — a* 
s *’*4 
4 - ax' 
4 “ 
. , m m\n m — 1 
Agam, let y — a + a- x x . 
into an infinite series, by dividing ax by a — * ; 
+ -^ + 
here we first have y — 
4- 
n 4" 1 
and 
X n 4- 1 
making y — 0, the latter part of the equation 
v? n 4 ~ 1 7 »« + m 
becomes = r ; whence tire 
I* X « -j- 1 
equation or fluent, properly corrected, is y == 
TT) n 4- 1 »2« 4 " m 
Hitherto a- and y 
+ ■ 
; but when y 
«4-i 
will 
y — 
4” 1 72 4” 1 
— A 
n -|- 1 
c' -j- l/x l Y X XX ; then, first, y = 
therefore, the fluent corrected, is 
c'+bx 2 ^ 
Si : 
y = 
- n + 
3 = Ay ; and the fluent of xyz -j-yxz 4" %a ~ 
xyx+ x y*±- 2 * L:= Zqx = But it sel- 
3 3 
dom happens that these kinds' of fluxions, which 
involve two variable quantities in one term, and 
vet admit of known and perfect fluents, are to 
be met with in practice. 
Having thus shewn the manner of finding 
such fluents as can be truly exhibited in alge- 
braic terms, it remains now to say something 
with regard to those other forms of expressions 
involving one variable quantity only; which 
vet are so affected by compound divisors and 
radical quantities, that their fluents cannot be 
accurately determined by any method whatso- 
ever. The only method with regard to these, of 
which there are innumerable kinds, is to find 
their fluents by approximation, which, by the 
method of infinite series, may be done to any 
degree of exactness. See the article Series. 
Thus, if it were proposed to find the fluent of 
— , it becomes necessary to throw the fluxion 
772 X « 4- 1 
are both supposed equal to nothing, at the same 
time ; which will not always be the case : thus, 
for instance, though the sine and tangent of an 
arch are both equal to nothing, when the arch 
itself is so ; yet the secant is then equal to the 
r«dius. It wall therefore be proper to add some 
examples, in which the value of y is equal to 
nothing, when that of a is equal to any given 
quantity a. Thus, let the equation y — x*x be 
proposed; whereof the fluent first found isy = 
thus, 
— . , xx , x x . XX 
X = AT -J j- 
X 
— T +, & c * Now the fluent of each term of 
this series, may be found by the foregoing rules 
X 2 y 3 v-t ,- l > 
to be x — J {- , _J — — — I — — 4-, &c. 
‘ U- 2 ‘ 4 a > ■ C-4 * ’ 
5a* 
Again, to approximate the fluent of 
X x*x 
we first find the value of 
f — a 2 ^ : 
, expressed in a series, to be (- 
2c 3 
1 , , 3a 1 1 
X X 2 -\ r — — — ■ r- X X* 4- 
2 ac 8 o' 4 ac 3 8 a s c ‘ 
5a 
1 6c‘ 
1 
1 6ac h 
16 
1 6a c 
- X * 6 +, &c. 
0, then = — , by the 
hypothesis; therefore the fluent, corrected, is 
A 3 ' — a* . . , , n_ 
y — — — . Agam, suppose^ — x x; then 
which value being multiplied by x x, and the 
fluent taken by the rules above laid down, we 
*4- 1 n 4“ 3 
get 
1 
72 -J- 1 X 
2c 5 
X 
3 a 
8 ?" 
2 ac n -j- 3 
n 4" 5 
4 * 
4 ac* 
Sa J c 
« + 
which, corrected, becomes 
And, lastly, if y — 
+ 
72+7 
v 
X TXT +’ 
3. To find the fluents of such fluxionary ex- 
pressions as involve two or more variable quan- 
tities, substitute, instead of such fluxion, its re- 
spective flowing quantity ; and, adding all the 
terms together, divide the sum by the number 
of terms, and the quotient will be the fluent. 
* . xy + xy 2xy 
Thus, the fluent of .vy -\-yx — — — — 
5a 3 11 
16? ~ L Sac' ' 1 6a V 16a' c 
&c. 
In order to shew the usefulness of fluxions, 
we shall give an example or two. Thus, suppose 
it were required, (1) To divide a given right 
line AB into two such parts, AC, CB, that their 
products or rectangles may be the greatest pos- 
sible. Let AB = a, and let the part AC, con- 
sidered as variable (by the motion of C towards 
B) be denoted by a-. Then BC being = a — a-, 
we have AC X BC = ax — xx, whose fluxion 
ax — 2 xx being put = 0, we get ax — 2xx ; 
and, consequently, a — \a. Hence it appears 
that AC (or a) must be exactly one-half of AB. 
(2) To find the fraction which shall exceed 
its cube by the greatest quantity possible : Let 
a- denote a variable quantity ; then the excess of 
x above a- 3 , being represented by a- — * 3 , if the 
fluxion of it be taken, we shall have x — 3a- 2 x 
— 0 ; therefore 1 — 3a- 2 , a 2 — j, and x — 
(3) To determine the greatest rectangle that 
can be inscribed in a given triangle. See Plate 
Miscel. fig. 91. Put the AC = b, and its altitude 
BD — a : let the altitude BS of the inscribed 
rectangle an, considered as variable, be denoted 
by x. Then, since AC and ac are parallel, it 
will be BD («) * AC ( b ) ] * DS (a — a) * — 
a 
— the line ac. And the area of the rectangle, or 
ac X BS = — - — , the fluxion of which is 
abx — 2b xx 
, and being put equal to 0, we have 
a — 2x, and x — 
Hence the greatest in- 
scribed triangle is that, the altitude of which is 
half the altitude of the triangle. 
(4) Of all right-angled plain triangles, con- 
taining the same given area, to find that of which 
the sum of the legs AB + BC is the least possi- 
ble. Let one leg AB be denoted by x, and the 
area of the triangle by a, then the other leg will 
be — , the fluxion of which is x — = 0 : 
FLY 
therefore ** = 2 a, and a = y's a. Whence BC 1 
= ( t ) 
\/ 2a 
~ + 2a. 
, ( 5 ) To determine the dimensions of the least 
isosceles triangle, ACD, fig. 92, that can circum- 
scribe a given circle. Let the distance QD of 
the vertex of the triangle f;om the centre of the 
circle be called x, and the remaining part OB. or 
radius, be represented by a ; then, if OS perpen- 
dicular to DC be drawn, we have DS =r ^/x 2 — a 2 : 
and since DC ; OS ( ; DB *, BC, we have BC = 
a X x — £7 
\/ A' 2 — a 2 
, which multiplied by BD or a + 
. a X A' + + 
gives • — — for the area of the triangle. 
v a — a 
Which being a minimum, its square is also a 
minimum, consequently or its equal 
: + + 
——^minimum .also; the fluxion of which 
3v X 
this being divided 
X_x — a — x X a- + ?1 3 
Ti 2 
by fL X J X +^ 2 
= o; 
, we get 
3 X X — a~ Ar + a=rO, whence 2,r — 4,7, and 
x — 2a. Therefore OD is equal to 20S, and 
DC ~ 2BC = AC; and so the triangle ACD, 
when the least possible, is equilateral. 
(6) Again, suppose it were required to find 
the solid contents of a spheroid, AFBH (plate 
Miscel. fig. 93). Let the axis AB, about which 
the solid is generated, be = a, the radius = p 
= 1, and the other axis FH of the generating 
ellipsis — b\ then, from the property of the el- 
lipsis, we have a 2 \ b 2 \ \ AD x BD (x x a — a ) 
y2 ' 
: DE 2 (y 2 ). Hence y 2 = — r x ax — xx: 
the fluxion of the solid s (— py 2 x) — 
X axx — x 2 x ; and the solidity j- — 
X \axx — jx s — the segment AIE ; which, 
when AD (,v) = AB (a), becomes (— - — x 
\ a 2 
i a> — \ a ") i — the content of the whole 
spheroid. Where, if b (FH) be taken := a (AB), 
we shall also get ±fia’’ for the true content of 
the sphere, whose diameter is a. Hence a sphere 
or spheroid is 4 of its circumscribing cylinder : 
for the area of the circle EH being expressed by 
— the content of the evlinder, whose diameter 
4 
is FA, and altitude AB, will be ; of which 
4 
i-pab 2 is evidently two-tliird parts. 
FLY, in zoology, a large order of insects, 
the distinguishing characteristic of which is, 
that their wings are transparent; by this 
they are distinguished from beetles, butter- 
flies, and grasshoppers. See Musca. 
Fly, in mechanics, a cross with leaden 
weights at its ends, or rather a heavy wheel, 
at right angles to the axis of a windlass, jack,. 
&c.; by means of which the force of the 
power, whatever it may be, is not only pre- 
served, but equally distributed in all parts of 
the revolution of the machine. See Me- 
chanics. 
The fly may be applied to several sorts of 
engines, whether moved, by men, horses, 
