FLU 
FLY 
that quantity, and thence the values of the 
others. 
Ex. 44 To inscribe the greatest paralle- 
logram D F G I in a given triangle ABC, 
fig. 10. , 
Draw B H perpendicular to A C ; put A C 
—a, BH=J,BE=r, thenEH=ft — x-, 
and by similar triangles, b: a ::x : DF= 
' "• 1 CtX 
hence, the area DFGI = — vl — x — 
max. or x x b ■ — x = b x — ■ x 2 — max, 
.\bx — 2xx = 0; hence, a- = 1 6 ; there- 
fore EH =iBH. 
2 
Ex. 5. Let ABC represent a cone, A C 
the diameter of the base ; to inscribe in it 
the greatest cylinder D FGI, fig. 11. 
Put p — , 78539, &c., then since A C 
p a 2 x 2 
— a . BH = b . BE = x — = 
the area of the end DEF of, the cy- 
linder; hence, the content of the cylin- 
der = ^ ~ b r ~ X ft — x = max. or x 2 x ft — x 
— b x 2 — x 3 — max. 2 b x x — 3 x 2 x 
2 1 
— o ; hence, x = - b ; therefore E H = - 
B H. See Cylinder. 
Ex. 6. To inscribe the greatest parallelo- 
gram D F G I in a given parabola ABC, 
fig. 11. 
Put BH = s, p = the parameter, x = 
B E ; then by the property of the parabo- 
la, DE'=p x, D E = p 2 x 1 , and D F 
i 1 
— ‘ip 2 x 2 ; hence, the area D F G I = 2 
j i — — — i 
pi x 1 x o — x — max. or x 2 x a — - x — 
ii It -i . 3i. 
ax 2 — x 2 = max. .•. - a x 2 x — - x- x 
2 2 
(l i 
= 0 ; hence, -=3a: 2 , or a = 3 x , x 
x 2 
1 2 
= - a ; consequently E H = -B H. 
O o 
Ex. 7. To cut the greatest parabola DEF 
from a given cone ABC, fig. 12. 
Let A G C be that diameter of the base 
which is perpendicular to D G F ; now E G is 
parallel to A B ; put A C = a, A B = b, C G 
= x, then AG — a — x; and by theproperty 
of the circle DG =\/ ax — f,.'.DF = 
2 \/ ax — * 2 ; also, by sim. As, a : ft :: x : 
h x 
G E = — : ; hence, we have the area of the 
a 1 
2 ^ ■ 
parabola = |X — X2 ax — x l — 
max. hence, x \/ ax — x 2 — max. or x 2 x 
a X ' — f = ai ! — f = max. 3 a x 2 x — 
3 
4i J i = 0, and 3a = 4r,.‘.j; = - a. See 
Simpson’s and Vince’s Fluxions. 
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 Entomology 
and Muse a. Flies are subdivided into 
those which have four, and those which 
have two wings. 
Fly, in mechanics, a cross with leaden 
weights at its ends, or rather a heavy 
wheel at right angles to the axis of a wind- 
las, jack, or the like ; by means of which 
the force of the power, whatever it be, is 
not only preserved, but equally distributed 
in all parts of the revolution of the ma- 
chine. 
The fly may be applied to several sorts 
of engines, whether moved by men, horses, 
wind, or water, or any other animate or 
inanimate power ; and is of great use in 
those parts of an engine which have a quick 
circular motion, and where the power of the 
resistance acts unequally in the different 
parts of a revolution. This has made some 
people imagine, that the fly adds a new 
power ; but though it may be truly said to 
facilitate the motion, by making it more uni- 
form, yet upon the whole it causes a loss of 
power, and not an increase : for as the fly 
has no motion of its own, it certainly re- 
quires a constant force to keep it in mo- 
tion; not to mention the friction of the 
pivots of the axis, and the resistance of the 
aii-. 
The reason, therefore, why the fly be- 
comes useful in many engines, is not that 
it adds a new force to them; but because, 
in cases where the power acts unequally, 
it serves as a moderator to make the mo- 
tion of revolution almost every where 
equal : for as the fly has accumulated in 
itself a great degree of power, which it 
equally and gradually exerts, and as equally 
and gradually receives, it makes the motion 
in all parts of the revolution pretty nearly 
equal and uniform. The, consequence of 
this is, that the engine becomes more easy 
and convenient to be acted on and moved 
by the impelling force ; and this is the only 
benefit obtained by the fly. 
The best form for a fly, is that of a heavy 
