In order to show the usefulness of luxions, we shall give an exam ,le or two, L. Suppose it were required to divide any riven right line A B into two such parts, , C, C B, that their products, or rec 'Ogles may be the greatest possible. eA B = a, and let tire part A C con ilde'ted as variable (bv the mot:on of C owa s B) be denoted by x. Then B C )eing a- x, we have AC X BC - ;xT x whose fluxion a — 2 x be ug put = , we get a a• -- 2 x ; and, :onsequently, x = a. thence it appears hat A C (or) must be exactly one half AB.
_Ex. 2. To &Nide a given number a nto two parts, x, k, so that xm yn may be maximum.
Since x+y -a, and xm yn - max. the luxion of each - 0, the former, because t is constant, and the !after, because it is maximum : 0, and m yn y = 0; hence, 1....-_ nacm n.ry d an m m y ;nxy herefore - or to y=n x, my nd m n x y. Now y a n x ma a, consequently x -}-n; nd 2ra m m-Fd If m=n, the two parts are equal.
Cor. Hence, to divide a quantity a nto three .parts, x, y, z, so that x y z may a max. the parts must be equal. For suppose s to remain constant, and y, z, to vary ; the product y ., and consequent ly x y z, will be the greatest when y = z. Or if y remain constant, the product x z, and consequently y x z, will be great est when x z. Thus it appears that the parts must be equal. And in like manner it may be shown, that whatever be the number of parts, they will be equal.
Ex. 3. Given x y z = a, and x z3 a maximum, to find x, y, z.
As x, y, z, must have some certain de terminate values to answer these condi tions, let us suppose such a value of y to remain constant, whilst .r and z vary till they answer the conditions, and then x -I- = 0 and z3 -I- 3 x z= 0; hence, 3 x i 3 x -i=- , z a = 3 x. Now let us suppose the value of z to remain constant, and x and y to vary, so as to satisfy the conditions ; then I: = 0, x+2xy y=0; hence, al: = - = 2 xi/ • y = 2 x ; substitute in the given equa tion, these values of y and z in terms of x, and x -I- 2 x± 3 x= a, or 6 x= a, 1 hence, x a; y 1 z a. In like manner, whatever be the number of unknown quantities, make any one of them variable with each of the rest, and the values of each in terms of that one quantity will be obtained; and by substi tuting the values of each in terms of that one, in the given equation, you will get the value of that quantity, and thence the values of the others. ,
Ex. 4. To inscribe the greatest paral lelogrim D G I, in a given triangle A B C, fig. 10.
Draw B H perpendicular to A C ; put A C = a, B II b,B E = x, then E If b - x; and by similar triangles, b : a the area DFGI a x 7-- —+ - = max. or x X b-x b x - -= max. b - 2 = 0 ; 1 1 hence, x - therefore E H H Ex. 5. Let A B C represent a cone, A C the diameter of the base to irt- %gibe in it the greatest cylinder D F fir. 11.
Put p'= 78539, &c. then since A C =a. 131I—b.BE—x the area of the end D E F of the linder; hence, the content of the cylin der = p a6' X b —x = max. or x' X b b — x3 max. 2 b x —3 x. a = 0 ; hence, x = — 2 b; therefore 3 B H = H. 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 bola, D E. = p x, D E =.pi xi, and D F= 2 pi xi ; hence, the area DFGI — 2 pi xi xa—x-- max. or A X —x--axi—A--max.
a — 3 —x 0; hence,— 3 X' or 2 1 3 x, x 5 a; consequently E H 2 == B. H.
3 Ex. 7. To cut the greatest parabola D E F from a given cone A B C, 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 a, A B b, C G x, then A G -- a — : and by the property of the circle D G ‘,/ a x—x. D F a x--x.; al so, by sim. a : b x : G E hence, we have the area of the parabola 2 b x 3X a — X 2../a x—x' = max. hence, x.,,/ a x—x. = max. or 5. X a x —x.
a x' — = max. 3 a x. — 4 x3 0, and 3 a = 4 x, x — 3 a. See 5 Simpson's and Vince's Fluxions.
FLY, in zoology, a large order of insects, or rather an indeterminate Iation, used to express a vast variety of insects belonging to different ordets. Entomologists apply the term only to in: dividuals of the genus Musca. See EN TOMOLOGY and 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, 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.