suppose x to remain constant, and y, z, to vary; the product y, and consequentlya y z, will be the greatest when y = = Or if y remain constant, the product x 2, and consequently y rz, 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 a y3 23 a maximum, to find x, y, z. As x, y, z, must have some certain determinate values to answer these condi. +5a+ tions, let us suppose such a value of y to In order to show the usefulness of fluxions, we shall give an example or two, 1. Suppose it were required to divide any given right line A B into two such parts, A C, C B, that their products, or rectangles may be the greatest possible. Let A B a, and let the part A C considered as variable (by the motion of C towards B) be denoted by r. Then B C being =Ɑ — x, we have ACX BC ax-xx, whose fluxion ax-2xx being put 0, we get a x 2 x x; and, consequently, x = a. Hence it appears that A ̊C (or ×) must be exactly one half of AB. remain constant, whilst r and z vary till they answer the conditions, and then x += 0 and 23 x + 3x z1 i = 0; hence, 3 x ziż 3 xi x 1 23 1 1 ... 2 = 3 x. Now let us suppose the value of 2 to remain constant, and x and y to vary, so as to satisfy the conditions; then + y 0, y'i + 2 x y ỳ = 0; 2 x y y 2xy hence, x=- y =y' v tion, these values of y and z in terms of 2, ..y=2x; substitute in the given equaand x + 2 x + 3 x = a, or 6 x = aj hence, x = a. In 6 a; :', y = za; z = 2 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 lelogram D F G I, in a given triangle A B C, fig. 10. Draw BH perpendicular to A C; put АС a, BH b, BE x, then E H b-x; and by similar triangles, b: « ax nx ; .. x b hence, the area D F GI and mnx: y. Now y If m=n, the two parts are equal. Cor. Hence, to divide a quantity a into three parts, x, y, z, so that x y z may be a max. the parts must be equal. For -x max. or x X b x-x2 - max. .. b x. 1 2 x x 0; hence, x= b; therefore E H Ex. 5. Let A B C represent a cone, A C the diameter of the base to in Ex. 7. To cut the greatest parabola DEF 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 G a, A B b, C G x, then A G a 2: and by the property of the circle D G ✔a x-x.. DF 2 u x-x2; also, by sim. ▲ 8, a : b :: x : G E hence, we have the area of the parabola 2 bx a bx ai max. hence, of insects belonging to different orders. Entomologists apply the term only to individuals of the genus Musca. See ENTOMOLOGY 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. 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 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 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 becomes 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 motion 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 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 every where equally distributed through the perimeter of the wheel, the motion will be more easy, uniform, and regular. In this form, the fly is most aptly applied to the perpendicular drill, which it likewise serves to keep upright by its centrifugal force: also to a wind lass or common winch, where the motion is quick; for in pulling upwards from the lower part, a person can exercise more power than in thrusting forward in the upper quarter, where, of course, part of his force would be lost, were it not accumulated and conserved in the equable motion of the fly. Hence, by this means, a man may work all day in drawing up a weight of 40%. whereas 3016, would create him more labour in a day without the fly. In order to calculate the force of the fly, joined to the screw for stamping the image upon coins, let us suppose the two arms of the fly to be each fifteen inches long, measuring from the centre of the weight to the axis of motion, the weights to be 50 pounds each, and the diameter of the axis pressing upon the dye to be one inch. If every stroke be made in half a second, and the weights describe an half circumference, which, in this case, will be four feet, the velocity will at the instant of the stroke be at the rate of eight feet in a second, so that the momentum of it will be 800; but the arms of the fly being as levers, each fifteen inches long, whilst the semi-axis is only half an inch, we must increase this force 30 times, which will give 24,000; an immense force, equal to 1006. falling 120 feet, or near two seconds in time; or to a body of 7506. falling 16 feet, or one second in time. Some engines, for coining crown-pieces, used to have the arms of the fly five times as long, and the weights twice as heavy, so that the effect is ten times greater. See COINING. FLY, in the sea language, that part of the mariner's compass on which the several winds or points are drawn. "Let fly the sheet," 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-amain, that it may hold no wind. FLY boat, a large vessel with a double prow, carrying from four to six hundred tons. FLYERS, in architecture, such stairs as go straight, and do not wind round; nor have the steps made tapering, but the fore and back part of each stair, and the ends, respectively, parallel to one another; so that if one flight do not carry you to your intended height, there is a broad half space, from whence you begin to fly again, with steps every where of the same length and breadth, as before. FLYING, the progressive motion of a bird, or other winged animal, in the li quid air. The parts of birds chiefly concerned in flying are the wings, by which they are sustained or wafted along. The tail, Messeurs Willoughby, Ray, and many others, imagine to be principally employ. ed in steering and turning the body in the air, as a rudder: but Boreli has put it beyond all doubt, that this is the least use of it, which is chiefly to assist the bird in its ascent and descent in the air; and to obviate the vacillations of the body and wings: for, as to turning to this or that side, it is performed by the wings, and inclinations of the body, and but very little by the help of the tail. The flying of a bird, in effect, is quite a different thing from the rowing of a vessel. Birds do not vibrate their wings towards the tail, as oars are struck towards the stern, but waft them downwards: nor does the tail of the bird cut the air at right angles, as the rudder does the water; but is disposed horizontally, and preserves the same situation what way so ever the bird turns. In effect, as a vessel is turned about on its centre of gravity to the right, by a brisk application of the oars to the left, so a bird, in beating the air with its right wing alone, towards the tail, will turn its fore part to the left. Thus pigeons, changing their course to the left, would labour it with their right wing, keeping the other almost at rest. Birds of a long neck alter their course by the inclinations of their head and neck, which altering the course of gravity, the bird will proceed in a new direction. The manner of flying is thus: the bird first bends his legs, and springs with a violent leap from the ground; then opens and expands the joints of his wings, so as to make a right line perpendicular to the sides of his body; thus the wings, with all the feathers therein, constitute one continued lamina. Being now raised a little above the horizon, and vibrating the wings with great force and velocity perpendicularly against the subject air, that fluid resists those succussions, both from its natural inactivity and elasticity, by means of which the whole body of the bird is protruded. The resistance the air makes to the withdrawing of the wings, and consequently the progress of the bird, will be so much the greater, as the waft or stroke of the fan of the wing is longer but as the force of the wing is continually diminished by this resistance, when the two forces come to be in equilibrio, the bird will remain suspended in the same place; for the bird only ascends FOC so long as the arch of air the wing describes makes a resistance equal to the excess of the specific gravity of the bird above the air. If the air, therefore, be so rare as to give way with the same velo. city as it is struck withal, there will be no resistance, and consequently the bird can never mount. Birds never fly upwards in a perpendicular line, but always in a parabola. In a direct ascent, the natural and artificial tendency would oppose and destroy each other, so that the progress would be very slow. In a direct descent, they would aid one another, so that the fall would be too precipitate. FLYING, artificial, that attempted by men, by the assistance of mechanics. The art of flying has been attempted by several persons in all ages. The Leucadians, out of superstition, are reported to have had a custom of precipitating a man from a high cliff into the sea, first fixing feathers, variously expanded, round his body, in order to break his fall. Friar Bacon, who lived near five hundred years ago, not only affirms the art of flying possible, but assures us, that he himself knew how to make an engine, wherein a man, sitting, might be able to convey himself through the air, like a bird; and further adds, that there was then one who had tried it with success; but this method, which consisted of a couple of large, thin, hollow copper globes, exhausted of the air, and sustaining a person who sat thereon, Dr. Hook shows to be impracticable. The philosophers of K. Charles the se cond's reign were mightily busied about this art. Bishop Wilkins was so confident of success in it, that he says, he does not question but, in future ages, it will be as usual to hear a man call for his wings, when he is going a journey, as it is now to call for his boots. The art of flying has, in some measure, been brought to bear in the construction and use of balloons. See ÆROSTATION, FLYING army, a small body, under a lieutenant or major general, sent to harass the country, intercept convoys, prevent the enemy's incursions, cover its own garrisons, and keep the enemy in continual alarm FLYING bridge. See BRIDGE. FLYING fish, a name given by the English writers to several species of fish, which, by means of their long fins, have a method of keeping themselves out of water some time. See ExоCOETUS, &c. FOCUS, in geometry and conic sections, is applied to certain points in the parabola, ellipsis, and hyperbola, where the rays reflected from all parts of these Foci of an ellipsis, are two points in the If from the foci two right lines are drawn, meeting one another in the periphery of the ellipsis, their sum will be always equal to the longest axis; and therefore, when an ellipsis and its two axes are given, and the foci are required, you need only take half the longest axis in your compasses, and setting one foot in the end of the shorter, the other foot will cut the longer in the focus required. Focus of an hyperbola, is that point in the axis, through which the latus rectum passes; from whence, if any two right lines are drawn, meeting in either of the opposite hyperbolas, their difference will be equal to the principal axis. See Hr PERBOLA. Focus of a parabola, a point in the axis within the figure, distant from the vertex one-fourth part of the latus rectum. PARABOLA. See Focus, in optics, is the point wherein rays are collected, after they have unSee dergone reflection or refraction. OPTICS. FODDER, any kind of meat for horses, or other cattle. In some places, hay and straw, mingled together, is peculiarly denominated fodder. FODDER, in mining, a measure containing twenty-two hundred and an half weight, though in London but twenty hundred weight. FOETUS, in anatomy, a term applied to the offspring of the human subject, or of animals, during its residence in the uteThe term of ovum is applied to the rus. fœtus, with its membranes and placenta taken altogether. We shall consider, under this article, the anatomy of the membranes which cover the fœtus during its abode in the uterus; of the placenta, which forms the medium of connexion between the systems of the mother and child; and of the pregnant uterus itself, since the peculiarities, distinguishing its structure this time, arise from the residence of the fœtus in its cavity. The following description applies to the uterus and its contents in the ninth month of gestation. The size of the organ differs much in different individuals; and this arises principally from varieties in the quantity of the liquor amnii. In shape it is oviform; the fundus answering to the largest extremity of the egg, and the cervix and os uteri to the small end. It The navel-string is a cord about two feet long, made of three vessels twisted together, and fixed at one end to the child's navel, at the other to the placenta. Its vessels are an umbilical vein and two arteries: the latter carry blood from the child to the placenta, and the former brings it back again. deviates from this regular figure from various accidental causes, as it adapts itself to the neighbouring parts, to the attitude of the body, and to the position of the contained child. Parts of the latter can often be distinguished in the living state. The small, or lower end of the uterus, is placed in the pelvis; this contains the greater part of the child's head, and fills Placenta. This, with the membranes, up the cavity so completely as to press makes a complete bag, lining the uterus, the bladder against the pubes, and the and containing the child, It is thick, rectum against the sacrum. The body fleshy, and exceedingly vascular. Its fiand fundus of the uterus, containing all gure is round and flat, about an inch the rest of the child and the placenta, is thick, and a span in breadth. The outer placed in the front of the abdomen, from surface, which adheres to the womb, is the pelvis upwards to the epigastric re- rough, tender, and bloody; the inner is gion, so as to be under and before all the smooth, harder, and marked by the rami other bowels. It occupies the whole fications of the vessels proceeding from space from one hip-bone to the other. the umbilical cord, which is attached to this part. Its substance consists of two parts intimately blended; viz. an umbili. cal, or infantine, and an uterine portion. The former is a continuation of the umbilical vessels of the fœtus, the latter an ef. florescence of the internal surface of the uterus. The fœtal portion, which is by far the largest part, is a regular ramification of the arteries and veins of the navelstring into smalier and smaller branches. No communication whatever has been discovered between these vessels and those of the uterus; so that the mode in which the foetus derives its nourishment and growth must be completely hidden from us. The round ligaments, Fallopian tubes, and ovaria, necessarily undergo consider able change in their situation: they be come closely connected to the uterus, as that body in its enlargement extends between the two layers of the broad ligaments. The ovaria are particularly distinguished after conception by containing a corpus luteum. This is a firm fleshy portion, distinguished by its yellowish gray colour from the rest of the ovary, and considered as a certain proof that conception has taken place. If there is one child, there is only one corpus luteum; if two children, two of these bodies, &c. The thickness of the pregnant uterus is from one to two-thirds of an inch. The arteries and veins of the uterus are wonderfully increased in size in the pregnant state, particularly opposite to the attachment of the placenta. This change seems to arise naturally from the important of fice which the vessels have to perform at this period; viz. the development and nutrition of the fœtus. Anatomists have disputed concerning the muscularity of the uterus; but Dr. Hunter describes the appearance of the muscular fibres, which are however very faint. The mouth of the uterus is closed, until the time of la bour, by a viscid glutinous substance. The contents of the pregnant uterus are, the secundines, liquor amnii, and the fatus. The former line the uterus, and immediately cover the child; they form the chain of connexion and communication between the bodies of the mother and child, and carry on that wonderful influence upon which the life and health of the child depend. They are divided into navel-string, placenta, and membranes; and, as they are expelled from the uterus after the birth of the child, they are called the after birth. The uterine portion of the placent a covers its convex surface in the form of a thin membrane, and detaches innumerable fine processes into the substance of the part. It seems to be a portion of the decidua. It is connected into one mass with the umbilical portion, and the ves sels of the uterus are continued into it, although they have no discoverable communication with the umbilical arteries and veins. The membranes are three in number; amnion, chorion, and decidua. The amnion is firm, thin, transparent, and possessing no visible vessels. It immediately includes the liquor amnii and child. The chorion lies outside of the amnion, and adheres to it; it is transparent, very thin and tender, and adheres externally to the decidua. The decidua is an efflorescence of the internal coat of the uterus, produced after conception, in order to adapt the womb for the ovum which is to enter it. It is shed after every birth, or miscarriage, with the other membranes; and hence its name. It is thicker, but more delicate and tender, than the amnion or |