FLUXIONS.

465

content of the cylinder, whose diameter is FA, and altitude AH, 1pb l a

will be--; of which $ p a b* is evidently the two-third parts.

4

Pros. IV. To compute the Surface of any Solid Body.(47.) The fluxion of the surface of the solid is equal to the peri-phery of the surface, by whose motion the solid is generated, mul-tiplied by its velocity on the edge of the solid, and the computa-tion is made as in the lbregoing.

Examp. i. Let it be proposed to determine the convex super-ficies ot a cone ABC, tig. 20, then the semidiameter of the base(B D or C D) being put = b, the slanting line, or hy pothenuse, AC— r, and FH (parallel to DC) = y, All = z, the surface = tr,its fluxion = w, andp = the periphery of a circle, whose diameter isunity, we shall, from the similarity of the triangles ADC and

O'

II m h, have b:c : \y (tit b ) . z (II A) = —: whence za (2pya)

b

Spcyji - pcy *

e=-; and consequently vs> — -. This, when y — k, be-

b b

comes = p c h =p x DC X AC = the convex superficies of th •whole cone ABC : which therefore is equal to a rectangle underhalf the circumference of the base and the slanting line.

Examp. 2. To find the surface generated bv the cycloidal curve.Put BF,, fig. 22, = z, FD = u, BF = x, 6D =a, s = the sur-face, and u = 6.28318, &c. then from the property of the cycloid

i _ i. . i — i

it will appear that & = a z x z x ; therefore s —pyx, — pya-x z x

- L _A. J-_ t i A

— p x a — x x a 2 x z x = p a z x z .i -— p a z x z z ; heuce s

= 2 pti 1 r z — f p a z x^, the surface generated by BE,which needsno correction, since when s = o, x — o\ when x — a, we haves — 4 p a?

-, equal to the whole surface generated by BA.

3

(48.) The method effluxions is also applied to find the centresof gravities, and oscillation of different bodies; to determine thepaths described by projectiles acted on by central forces, the re-sistances to motions in resisting media, tiie attractions of bodiesunder different forms, the directions of wand which has the greatesteffect on an engine, and to solve many curious and useful problems,as may he seen in the authors on this subject above mentioned. Thetreatise by Simpson, and that hy Vince, deserve particular notice.

FLY, in entomology, a large order of insects, the distinguishing•haracterislic of which is, that their wings are transparent. By thisthey are distinguished from beetles, butterflies, grasshoppers’, &c.Flies are subdivided into those which have four and those whichhave two wings. See Apis, Formica, Culex, Muse a, &c.

Fly, in mechanics, a cross with leaden weights at its ends; orrather a heavy wheel at right angles, to the axis of a windlass, jack,or the like ; by means of which, the force of the power, whateverit is, is not only preserved, but equally distributed in all parts ofthe revolution of the machine. See Mechanics. The fly maybe 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 quickcircular motion, and where the power of the resistance acts une-qually in the different parts of a revolution. This has made somepeople imagine, that the fly adds a new power; but though it maybe truly said to facilitate the motion, by making it more uniform,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 requires a con-stant force to keep it in motion ; not to mention the friction of thepivots of the axis, and the resistance of the air. The reason, there-fore, why the fly becomes useful in many engines, is not that it !Reids a new force to them ; but because, in cases where the power iacts unequally, it serves as a moderator to make the motion of re- |volution almost every where equal: for as the fly has accumulated jin itself a great degree of power, which it equally and graduallyexerts, and as equally and gradually receives, it makes the mo- !tion in all parts of the revolution pretty nearly equal anduniform. The very manifest consequence "of this’is, that theengine becomes more easy and convenient to be acted on andmoved by the impelling force ; and this is t he only benefit obtainedby the fly. The best form for a ily, is tjiat of a heavy wheel orcircle, of a fit size, as this will not only meet with less resistancefrom the air, but being continuous, and the weight every whereequally distributed through the perimeter of the wheel, the motionwill he more easy, uniform, and regular. In this form, the fly ismost aptly applied to t he perpendicular drill, which it likewise servesfo keep upright by its centrifugal force : also to a windlass or com-mon winch, where the motion is quick; for in pulling upwardsfrom tin: lower part, a person can exercise more power than inthrusting forward in the upper quarter: where, of course, part of;bis force would be lost, were it not accumulated and conserved in ijthe equable motion of the fly. Hence, by this means, a man maywork all day in drawing up a weight of 40lb. whereas 30!b.would create him more labour in a day without the fly.

Fly, in the sea language, that part of the mariner's compass onwinch the several winds or points are drawn. “ Let fly the sheet,” \vol. n.—xo. 86.

is a word of command to let loose the sheet, in case of a gust of. wind, lest the ship should overset, or spend her top-sails and masts;which is prevented by letting the sheet go a-main, that it mayhold no wind.

Fly-Boat, a large vessel with a double brow, carrying from fourj to six hundred tons.

] FLYERS, in architecture, such stairs as go straight, and do not1 wind round ; nor have the steps made tapering, but the fore andback part of each stair, and the ends, respectively parallel to oneanother; so that if one flight do not carry you {o your intendedheight, tiiere is a broad half space, from whence you begin to flywith steps every where of the same length and breadth, as before.

Fly-Fishing, the act, or art, of catching fish with a fly.The fly with which the hook is baited may be either natural or ar-tificial ; they are made of various sizes and colours according to thesort of fish or the season. See Fishing.

I’ly Honeysuckle. See Lonicera , No. 10.

Fly Honeysuckle, African. See Halleria.

FLYING, the progressive motion of a bird, or other wingedanimal in the air. The parts of birds chiefly concerned in flyingare tire wings and the tail: by the former, the bird sustains’andwafts himself along; and by the latter, he is assisted in ascendingand descending, to keep his body poised and upright, and to ob-viate the vacillations thereof. It is by the largeness and strengthof the pectoral muscles, that birds are so well disposed for quick,strong, and continued flying. These muscles, which in men arescarce a 70th part of the muscles of the body, in birds exceed-andoutweigh all the other muscles taken together; upon which Mr.Willoughby remarks, that if it be possible for a man to fly, hiswings must be so contrived and adapted, that lie may use bi’s legs,and not his arms, in managing them. The tail, Messrs. Willough-by, Bay, and many others, imagined"to be principally employedin steering and turning the body, as a rudder; but Borelli has putit beyond all doubt, that this is the least use of it. Its chief use isto assist tiie bird in its ascent and descent in the air, and to obvi-ate t^ie vacillations of the body and wings; for, as to turning.thebody to this or that side, it is performed by the wings and incli-nation of the body, and but very little by the help of tin* tail.The flying of a bird, in fact, is quite a different thing from the row-ing of a vessel. Birds do not vibrate their wings towards the tailas oars are struck towards tiie stern, but waft them downwards; nordoes tiie tail of the bird cut tiie air at right angles, as the rudderdoes the water; hut is disposed horizontally, and preserve; thesame situation what way soever the bird turns, in efleet, as avessel is turned about on its centre of gravity to tiie right, by abrisk application of the oars to the left; so a bird in healing theair with its right wing alone, towards tiie tail, will turn its fore-part

6C to