FLY 



;ribe in it the greatest cylinder D F G I, 

 Put /> = 78539, &c. then since A C 



the area of the end D E F of the cy- 

 linder; hence, the content of the cylin- 



X b x = max. or x 1 



dei _ f * 



X b x = b x 1 - x3 = max. .-. 2 b x x 



2 

 3 x* x = ; hence, x = ~b; therefore 



B H = - B H. 



o 



See CYLINDER. 



Ex. 6. To inscribe the greatest paral- 

 lelogram D F G I in a given parabola 

 A B C, fig. 11. 



Put B H = a, p = the parameter, x = 

 B E ; then, by the property of the para- 

 bola, D E J == p x, :. D E =.p% x$, and 

 D F = 2 p\ xl ; hence, the area D F G I 

 = 2 pi xi x a x = max. or x$ X 



i 3 1 i 



x = ax? jcz = max. /. - a x 2 x 



~x x = 0; hence,- = 4 oS %, or a 



* 2 



x 



3 07, .*. x = - a; consequently E H 



-fa,* 



Ex. 7. To cut the greatest parabola 

 D E F 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 G = 

 a, A B = b, C G == x, then A G = a 

 x: and by the property of the circle D G 

 = v/ a x x 1 .-. D F = 2v/ a x x 1 ; al- 

 so, by sim. A *, a : b :: x : G E = 



a 5 

 hence, we have the area of the parabola 



2 bx 



= ^X X 2^/axx 3 - = max. hence, 



6 a 



x^/ a x x* = max. or x* X a J' -j- 1 = 

 a x 1 x* =*= max. .'. 3 a x 1 x 4 x3 x 



= 0, and 3 a = 4 ,r, /. x a. See 

 5 



Simpson's and Vince's Fluxions. 



FLY. in zoology, a large order of 

 insects, or miner an indeterminate ap- 

 pellation, used to express a vast variety 



FLY 



of insects belonging to different orders. 

 Entomologists apply the term only to in- 

 dividuals of the genus Musca. See EN- 

 TOMOLOGY and Mi SCA. 



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, 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 machine. 



The fl may be applied to several sorts 

 of engines, whether moved by men, hor- 

 ses, wind > or water, or uny 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 uniform, yet upon the 

 whole it causes a loss of power, and not 

 an increase : for as the fly has no mo- 

 tion of its own it certainly requires a 

 constant force to keep it in motion ; 

 not to mention the friction of the pivots 

 of the axis, and the resistance of the 

 air. 



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 obtain- 

 ed by the fly. 



The best form for a fly, is that of a 

 heavy wheel or circle, of a fit size, as this 

 will not only meet with less resistance 

 from the air, but being continuous, and 

 the weight even where equally distribut- 

 ed through the perimeter of the wheel, 

 the motion will be more easy, uniform, 

 and regular. In this form, the fly is most 

 aptly appl-ed to the perpendicular drill, 

 which it likewise serves to keep upright 

 by its centrifugal force : also to a wind- 





