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we have sin z=;
=z, and cos z=1; and therefore
sin (z+ z) = sin z + z cos z; whence sin (x + z) —
sin z, or (sin z)⋅ = z c
=z cos z: viz. the fluxion of the
sine of an arc whose radius is unity, is equal to
the product of the fluxion of the angle into the
cosine of the same arc.

In like manner the fluxion of cos z, or cos
(+)-cos z = cos z cos z— sin z sin x cos z,
since(art.SINE)cos(3 + 2) = = cos z cos z- sin z sin z:
therefore, because sin x=x, and cos z=
z=1, we
have (cos z) = cos - sin - cos z=-ż sin z:
that is, the fluxion of the cosine of an arc, radius
being 1, is found by multiplying the fluxion of
the arc (taken with a contrary sign) by the sine

of the same arc.

By means of these two formulæ, many other fluxional expressions may be found. As that (cosm z)· = •——m z sin mz. (sin mx)' = + m z cos m z. (tan 2) = (3

(cotan z):

(sec z) =

sin z

COS 3

=

sin 2 x

z sin z

COS 25

% COS %
sin 2g

log of ; hence z=zrx log of e=eix log of e,
the fluxion of the proposed exponential ex; and
which therefore is equal to the said proposed (cosec =).
quantity, drawn into the fluxion of the exponent,
and also into the log. of the root.
Also in the second case, put the exponential (sin ":)=m
=z; then the logarithms give log. z=1x log. y, (cos ms).: =-m cOS

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=m sin

2

(cos22-sin2)
COS2%

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* sin 2.

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XI. To find the second, third, c. fuxion of flowing quantity.-These fluxions differ in nothing, except their order and notation, from first fluxions, being actually such to the quantities from which they are immediately derived; and therefore, they may be found, in the same manner, by the general rules already delivered.

X

Thus, by the 4th rule, the first fluxion of x3 is 3xx; and if x be supposed constant, or if the root a be generated with an equable celerity, the fluxion of 3x2x, or 3xxx2, will be 3x × 211=6xx2, which is the second fluxion of x3; and 613 will be its third fluxion: but if the celerity with which x is generated be variable, either increasing or decreasing, then x being variable, will have its fluxion denoted by x, &c. In this case the fluxion of 3xxx will be, by the 2d and 4th rules, 6xxx i+ 3x2×x=6xx2+3121, the second fluxion of x3. And the third fluxion of a3 obtained in like manner from the last, will be 6x x x2 + 6x × 2x x + 6xx × x + 3x2x=6x3 + 18xxx+Sa2. Thus also, if 1-9 12 + 1 y=nx a then y=nXn-1 x s then 22% = xy +

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X. To find the fuxions of sines, cosines, &c. Suppose we require the fluxion of sin z, that is, the sine of the angle or are denoted by z, we must suppose that by a motion of one of the legs including the angle, it becomes z +z, then sin (≈ + z) sin z is the fluxion of sin z. But according to the formula for the sines of sums of arcs (see SINE and TRIGONOMETRY), we have sin (≈ + z) = sin z cos 2+ sin x cos z the radius being assumed equal to unity. But the sine of an arc indefinitely small does not differ sensibly from that are itself, mor its cosine differ perceptibly from radius: hence, to obtain the second fluxion it will suffice t●

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If the function proposed were ar, we should find (ax)= ': the factors na and x being regarded as constant in the first fluxion nax

11

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(ax) en (n-1) (n-2) (n-3) ax" x+, &c. And the fluxional coefficients will have the following values:

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(axn)..

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=nax

—=n (n−1) ax

(axn):

(axn)

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- = n (n−1) (n−2) az "

1113

N-4

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of (a3 + 23)42% is ‚¦ (a3 +23)5.

III. When the root under a vinculum is a compound quantity; and the index of the part or factor without the vinculum increased by 1, is some multiple of that under the vinculum.-Put a single variable letter for the compound root; and substitute its powers and fluxion instead of those, of the same value, in the given quantity; so will it be reduced to a simpler form, to which the preceding rule can then be applied.

So, if the given fluxion be F= (a2 + ×) 3×3x; = n (n−1) (n−2) (n−3) ax &c. where 3, the index of the quantity without the vinculum, increased by 1, makes 4, which is double of 2, the exponent of x2 within the same; therefore putting a2+x2, thence 2=s—a2, and its fluxion is 2xx=z; hence then x3x=&x= (2-a2), and the given quantity For (42+x2)3x3 is = § 2a x(x—a2) o

Hence it is manifest that in the case where the exponent n is a positive whole number, the function ar" has only a limited number of fluxions, of which the most elevated is (•)n ax"=n (n-1) (n−2)....... 2.1.ax”; an expression which is no longer susceptible of fluxing, since it contains no more variable quantities.

Inverse method of fluxions, relates to the finding of fluents, and is of great utility though of considerable difficulty.

As it is only in certain particular forms and cases that the fluents of given fluxions can be found; there being no method of performing this universally a priori, by a direct investigation; like finding the fluxion of a given fluent quantity; we can do little more than lay down a few rules for such forms of fluxions as are known, from the direct method, to belong to such and such kinds of fluents or flowing quantities; and these rules, it is evident, must chiefly consist in performing such operations as are the reverse of those by which the fluxions are found to given flowing quantities. The principal cases of which are as follow:

1. To find the fluent of a simple fluxion; or that in which there is no variable quantity, and only one Aurional quantity. This is done by barely substituting the variable or flowing quantity instead of its fluxion, and is the result or reverse of the notation only. Thus,

The fluent of ax is ax.

The fluent of ay + 2y is ay +2y.
The fluent of Va2 + x2 is √ a2+x3.

II. When any power of a flowing quantity is multiplied by the fluxion of the root. Then, having substituted, as before, the flowing quantity for its fluxion, divide the result by the new index of the power. Or, which is the same thing, take out, or divide by, the fluxion of the root; add 1 to the index of the power; and divide by the index so increased.

So if the fluxion proposed be
Strike out x then it is

3r3x;

or =

5

a2; and the fluent of each term gives F-23:13 (¿s—¡¡a); or, by substituting the value of instead of it, the same fluent is 3(a2+x2) } × (‚}x2—¿¿«2), or †}, a2 + xa`} x2—¿a2.

IV. When there are several terms involving two or multiplied by the other quantity or quantities.— more variable quantities, having the fluxion of each Take the fluent of each term, as if there was only one variable quantity in it, namely that whose fluxion is contained in it, supposing all the others to be constant in that term; then if the fluents of all the terms so found be the very same quantity, that quantity will be the fluent of the whole.

Thus, if the given fluxion be xy + xy. Then, the fluent of xy is xy, supposing y constant; and the fluent of zy is also xy, when is constant ; therefore the common resulting quantity xy is the required fluent of the given fluxion xy+xy. And in like manner the fluent of xy + xyz + xy is xyz.

V. When the given fluxional expression is in this

xy-xy form viz. a fraction including two quantiya ties, being the fluxion of the former drawn into the latter, minus the fluxion of the latter drawn into the former, and divided by the square of the latter : then the fluent is the fraction or of the former quantity divided by the latter. That is, xy-xy * the fluent of y

and the fluent of

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Though the examples of this case may be performed by the foregoing one. Thus the given

fluxion

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xy-xy
ул

y

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For a 2d ex. suppose it be proposed to find the when fluent of xy + xy. Here F=xy + xy; then, writing x for x, and y for y, it is xy + xy or 2xy=G; the fluxion of which is 2xy + 2xy=Ġ; then Ġ: F:: G: F becomes 2xy + 2xy : xy + xy : : 2xy : ay=F, the fluent sought.

is

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VIII. But fluents are more generally found by means of a table of forms of fluxions and fluents, in which comparing any proposed fluxion with a corresponding form given in the table, the fluent of it will be found. A very useful table of this kind is inserted under the article FLUENT in Dr. Hutton's Math. Dictionary, and illustrated by examples. But the most comprehensive table of this kind is from Cotes's Harmonia Mensurarum, and given in Emerson's Fluxions; its use has been amply shewn by Holliday and succeeding writers.

IX. To find fluents by means of infinite series.-When a finite form cannot be found to agree with a proposed fluxion, it is then usual to throw it into an infinite series, either by division, or extraction of roots, or by the binomial theorem, &c. after which the fluents of all the terms are taken separately.

For ex. To find the fluent of

1-x
1+x-x2

Here, by dividing the numerator by the denomi nator, this becomes x-2xx + 3x2x- 5x3x + 8x+x, &c.; and the fluents of all the terms being taken, give x− x2 + x3— 2x + 3x3, &c., for the fluent

, is hyp. log. (x±a+ √x2±2ax); sought.

hyp, log. of

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a-x

is hyp. log. of

a- Q2味 x 2

a+ √ a2±x2 VIL Many fluents may be found by the direct method of Auxions, thus.-Take the fluxion again of the given fluxional expression, or the 2d fluxion x2

of the fluent sought; into which substitute

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also y,y, y, &c. in continual proportion. Then divide the square of the given fluxional expression by the 2d fluxion, just found, and the quotient will be the fluent sought in many cases.

Or the same rule may be delivered thus. In the given fluxion F write x for x, y for y, &c. and call the result G, taking also the fluxion of this quantity, Ġ; then make G: F:: G: F, so shall the fourth proportional F be the fluent, as before. This is the rule of M. Paccassi.

Some excellent examples of this method may be seen in Simpson's Fluxions, vol. i.

To correct a Fluent.-The fluent of a given fluxion, found as above, sometimes wants a correction, to make it contemporary with that required by the problem under consideration, &c.: for the fluent of any given fluxion, as x, may be either a (which is found by the rule) or may be i±c, that is x plus or minus some constant quantity ; because both x and xc have the same fluxion : and the finding of the constant quantity, is called correcting the fluent. Now this correction is to be determined from the nature of the problem in hand, by which we come to know the relation which the fluent quantities have to each other at some certain point or time. Reduce therefore the general fluential equation, found by the rules above, to that point or time; then if the equation be true at that point, it is correct; but if not, it wants a correction, and the quantity of that corsides of the equation when reduced to that parrection is the difference between the two general ticular state. Hence the general rule for the correction is this:

Connect the constant, but indeterminate, quantity with one side of the fluential equation, as It may be proved whether this be the true determined by the foregoing rules; then, in this fluent, by taking the fluxion of it again, which, equation, substitute for the variable quantities if it agree with the proposed fluxion, will shew such values as they are known to have at any parthat the fluent is right; otherwise, the method ticular state, place, or time; and then from that fails. particular state of the equation find the value of Thus, if it be proposed to find the fluent of, the constant quantity of the correction. 21i. Here F = nx”− 1¢ *; write first x for x, Ex. To find the correct fluent of gar3x. First

the general fluent of this is z=a14, or z=ax++c, taking in the correction c.

2

Now if it be known that and r begin together, or that z = (0, when r = 0; then writing 0 both for * and z, the general equation becomes () = 0 + c, or c=0; so that the value of c being 0, the correct fluents are z=axt.

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ab + c

from

But if z be =0, when r is=b, any known quantity; then substituting 0 for z, and b for x, in the general equation, it becomes 0 which is found c-ab1; and this being written for it in the general equation, this becomes z= a14ab4, for the correct or contemporary fluents.

Or lastly, if it be known that z is some quantity d, when a is equal some other quantity, as b; then substituting d for z, and b for 1, in the general fluential equation x = an4 + c, it becomes d= ab + c; and hence is deduced the value of the correction, viz. c = d - ab4; consequently, writing this value for in the general equation, it becomes z =a14 - ab+d, for the correct equation of the fluents in this case.

And hence arises another easy and general way of correcting the fluents, which is this: In the general equation of the fluents, write the particular values of the quantities which they are known to have at any certain time; then subtract the sides of the resulting particular equation, from the corresponding sides of the general one, and the remainders will give the correct equation of the fluents sought. So, as above, the general equation being write d for 2, and b for 1, then hence by subtraction

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z=ax4;

d= alt ;

z-d=a4 - ab4, or z=ax^—aba + d, the correct fluents as before. (Hutton's Dict.) As this is a very important part of the doctrine of fluxions, it may be as well to illustrate it a little farther. Let us demand, therefore, the area of a parabola, commencing not at the summit A, but at the point B (fig. 11. Pl. 68.), taken upon the axis at any distance whatever from that summit. Here, then, the abscissa BP being represented by a, the distance AB from the vertex by a, and the parameter by p, we have for the equation of the curve, y = V pa + px = PQ ; and the fluxion of the area will be yx = x y pa + px : the fluent of this found by the usual methods will be

apa + pa pa + pr

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= 3(a + x)√√ pa + px

AP PQ. Now, it is evident that the area required should be nothing, when a or BP becomes nothing. The area above is therefore too great by the area ABC; and this area ABC is found by making x-0 in the above expression; for it then becomes tapa=AB BC. Whence we see that to apply this correction to the fluent, it is requisite to know what it becomes when x is made equal to nothing; and if the result be positive, it must be subtracted; but if it be negative it must be added: for in the former case the fluent is too great, and in the latter it is too little.

There are cases also where by the conditions of the problem, we know that the fluent ought to be equal to nothing when the abscissa or x, instead of being 0, is of a determinate magnitude as a. It is then requisite to suppose r in the fluent equal to a; and if the result is positive to subtract it from the fluent, since it is too great; and, on the contrary, to add it if the result is negative, since it is too little. This would have happened if, in

the preceding problem, the abscissa AP had been supposed equal to x, while we only required the magnitude of the segment CBPQ. For we should find the fluxion of this area =x/pr, and its fluent =xpr. But this expression is that of the total area APQ, and the area demanded should vanish when AP=AB. We must therefore suppose r = , and the preceding fluent becomes reduced to apa, which being deducted from the former expression, will give for the true value of the area sought, rpr-apa

A sketch of the principles of fluxions being now delivered, we may next say a little respecting the chief writings and improvements that have been made by divers authors, since the first discovery of them: indeed some of the chief improvements may be learned by consulting the preface to Dr. Waring's Meditationes Analytica. fluxions to a considerable degree of perfection; as The inventor himself brought the doctrine of may be seen by many specimens of this science, his Tract on Quadratures, and in his Treatise on given by him; particularly in his Principia, in Fluxions, published by Mr. Colson; from all of which it will appear, that he not only laid down the whole theory of this method, both direct and inverse; but also applied it in practice to the solution of many of the most useful and important problems in mathematics and philosophy.

Various improvements however have been made by many illustrious authors on this science; particularly by John Bernoulli, who treated of the fluents belonging to the fluxions of exponential expressions; James Bernoulli, Craig, Cheyne, Cotes, Manfredi, Riccati, Taylor, Fagnanus, ClaiLe Grange, Emerson, Simpson, Landen, Waring, raut, D'Alembert, Euler, Condorcet, Walmesley, Bezout, Bossut, Lacroix, &c. There are several other treatises also on the principles of Fluxions, by Hayes, Newyentyt, L'Hôpital, Hodgson, Rowe, Vince, &c. &c., delivering the elements of this science in an easy and familiar manner. been delivered by its great author in so concise a manner, as to give occasion to the most ingenious bishop Berkeley to represent it as founded on inThis author, in a letter under the title of the conceivable principles, and full of false reasonings. Analyst, published in the year 1734, has been at great pains to convince his readers, that the objects, principles, and inferences of the modern analysis by fluxions, are not more distinctly conceived, or more evidently deduced, than religious mysteries and points of faith. He says he does not controvert the truth of the conclusions, but only the logic and method of mathematicians. An answer to the objections of the Analyst appeared very early under the name of Philalethes Cantabrigiensis, since known to be Dr. Jurin. And various others were published on the same side of the question. But the most able defences of the doctrine of fluxions were made by Mr. Robins (see his Tracts, vol. ii.) and by Mr. Maclaurin, in his very elaborate and excellent Treatise on Fluxions. The demonstrations of these two gentlemen are, we think, sufficient to satisfy the most scrupulous: to them, therefore, we with pleasure refer. Mr. Woodhouse, who, in his Principles of Analytical Calculation, has revived many of Berkeley's objections with greater mathematical talent, but with less metaphysical acuteness, says, the prolixity of the reasonings of Maclaurin and Robins "confirm the notion that the

The elements of the doctrine of fluxions have

method they defend is an incommodious one." This remark we think uncandid: the prolixity is occasioned not by want of simplicity in the method they defend, but by the tortuous intricacy and obscurity of many of the objections, which required much time to be pursued throughout, and, as several of them were subtile and specious, much labour and room to refute them completely. With regard to the comparative perspicuity of the two methods of establishing the principles, from the doctrine of motion and from pure analysis, we cannot hesitate to say that the former has the preference: and we are convinced that every unbiassed reader of the treatises of Maclaurin and Simpson on the one hand, and of Bossut and Lacroix on the other, will agree with us in opinion. Yet, we are constrained to acknowledge, notwithstanding this, that the principal improvements and extensions of the modern analysis, during the last forty years, have been made by continental, not by British, mathematicians.

For the various applications of fluxions to practical purposes, see the articles ASYMPTOTE, INFLEXION, MAXIMA, TANGENT, &c.

To FLY. v. n. pret. flew or fled; part. fled or flown: fled is properly from flee. (Fleogan.) 1. To move through the air with wings (Shak speare). 2. To pass through the air (Job). 3. To pass away (Prior). 4. To pass swiftly (Pope). 5. To move with rapidity (Dryden). 6. To part with violence (Shakspeare). 7. To break; to shiver; to burst asunder with a sudden explosion (Swift). 8. To run away; to flee (Prior). 9. To FLY at. To spring with violence upon; to fall on suddenly (South). 10. To FLY in the face. To insult (Swift). 11. To FLY in the face. To act in defiance. 12. To FLY off. To revolt (Addison). 13. TO FLY out. To burst into passion (Ben Jonson). 14. To FLY out. To break out into licence. 15. To FLY out. To start violently from any direction (Bentley). 16. To let FLY. To discharge (Glanville).

To FLY. v. a. 1. To shun; to avoid; to decline (Shakspeare). 2. To refuse association with (Dryden). 3. To quit by flight (Dryden). 4. To attack by a bird of prey (Bacon).

FLV, in entomology. See MUSCA.
FLY (Honeysuckle), in botany. See LOXI-

CERA.

FLV (Honeysuckle), African. See HAL

LERIA.

FLY-CATCHER, in ornithology. See Mus

CICAPA.

FLY (Spanish). See CANTHARIDES. FLY-TRAP (Venus's), See DIONEA. FLY THE HEELS, in the manage, is when a horse obeys the spurs.

FLY (Vegetable), a very curious natural production, chiefly found in the West Indies, and thus improperly denominated. Excepting that it has no wings, it resembles the drone both in size and colour more than any other British insect. In the month of May it buries itself in the earth, and begins to vegetate. By the latter end of July, the tree is arrived at its full growth, and resembles a coral branch; and is about three inches high, and bears several little pods, which dropping YOL. V.

off become worms, and thence flies, like the British caterpillar. Such was the account originally given of this extraordinary production. But several boxes of these flies having been sent to Dr. Hill for examination, his report was this: "There is in Martinique a fungus of the clavaria kind, different in species from those hitherto known. It produces soboles from its sides; I call it, therefore, clavaria sobolifera. It grows on putrid animal bodies, as our fungus ex pede equino from the dead horse's hoof. The cicada is common in Martinique, and in its nympha state, in which the old authors call it tettigometra: it buries itself under dead leaves to wait its change; and, when the season is unfavourable, many perish. The seeds of the clavaria find a proper bed in this dead the cicada in the British museum; the clavainsect, and grow The tettigometra is among ria is just now known. This is the fact, and all the fact; though the untaught inhabitants suppose a fly to vegetate, and though there is a Spanish drawing of the plants growing into a trifoliate tree, and it has been figured with the creature flying with this tree upon its back." Edwards has taken notice of this extraordinary production in his Gleanings of Natural Hisrtoy.

FLY, in mechanics, is a name given to a certain appendage to many machines, either as a regulator of their motions, or as a collector of power. In the first case the fly is a heavy disk or hoop, or other mass of matter balanced on its axis, and so connected with the machinery as to turn briskly round with it. This may be done with the view of rendering the motion of the whole more of the accelerating forces, or of the resistances regular, notwithstanding unavoidable inequalities occasioned by the work. It becomes a REGULATOR. Suppose the resistance extremely unequal, and the impelling power perfectly constant; as when a bucket wheel is employed to work one pump. When the piston has ended its working stroke, and while it is going down the barrel, the power of the wheel being scarcely opposed, it accelerates the whole machine, and the piston arable velocity. But in the rising again, the wheel is opposed by the columm of water now pressing wheel; and when the piston has reached the top of on the piston. This immediately retards the

rives at the bottom of the barrel with a consider

the barrel, all the acceleration is undone, and is to begin again. The motion of such a machine is very hobbling: but the superplus of accelerating force at the beginning of a returning stroke will not make such a change in the motion of the machine if we connect the fly with it. For the accelerating momentum is a determinate quantity, Therefore, if the radius of the fly be great, this momentum will be attained by communicating a small angular motion to the machine. The momentum of the fly is as the square of its radius; therefore it resists acceleration in this proportion; and although the overplus of power generates the same momentum of rotation in the whole machine as before, it makes but a small addition to its velocity. If the diameter of the fly be doubled, the augmentation of rotation will be reduced to one-fourth. Thus, by giving a rapid motion to a small quantity of matter, the great acceleration, during the returning stroke of the piston, is prevented, This acceleration continues, however,

C

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