(tan :)=( COS 3 cos? 2 sinx cosa Z COS 5 . sin ? : COS 2. 3013 +1242? + ea'r + 25a* + caorxc we have sin z=2, and cos z=1; and therefore = the flur1'11 sin (z+2) = sin z+ż cos z; whence sin (2+i)ion of the given quantity. sin z, or (sin z)•=z cos z: viz. the fluxion of the VII. To find the fiuxion of a logarithm. —The fluxion sine of an arc whose radius is unity, is equal to of the hyperbolic logarithm of any quantity is the product of the fluxion of the angle into the equal to the flusion of that quantity divided by cosine of the same arc. the quantity itself: e. gr. the fluxion of the hyper- In like manner the fluxion of cos z, or cos bolic logarithm of = is = 5: the flution of the (2+2), cos z= cos z cos 3–sin 2 sin z - cos , since(art. Sine)cos(: +z)=cos z cos — sin sin 2: therefore, because sin ż=2, and cos ż=1, we common logarithm of 1 (viz. will be L have (cos z). = cos :-; sin :- cos s=-ż sin z: that is, the fluxion of the casine of an arc, radius --XM; or, the fluxion of the hyperbolic loga- being 1, is found by multiplying the fiuxion of rithm of any number multiplied by M or 0-43429, the arc (taken with a contrary sign) by the sine &c. is = the fuxion of the common logarithm of of the same arc. the said number. By means of these two formulæ, many other VIII. To find the flution of exponential quantities, tbar fluxional expressions may be found.' As that is, quantities which have ibeir esponent a flowing or (cos mz):=-mz sin mz. sariabie letter. These are of two kinds, viz. when the root is a constant quantity, as ex; and when the (sin mz) = +m z cos mz. root is variable, as yr. sin z z (cos? 2-sin's) In the former case, put the proposed exponen cos? tial ex=z, a single variable quantity; then take the logarithm of each, so shall log of z=xx log. (cotan z) = of e; take the fiuxions of these, so shall i sin (sec z): log of e; hence :=zi x log of ezeti x log of en the iuxion of the proposed exponentialer; 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 sin g'=z; then the logarithms give log. Z=1x log. y, (cos m 3): XI. To find the second, third, &c. Maurion of a and the fluzions give =* * log.+*X y frowing quantity. These fluxions differ in nothing, except their order and notation, from first fluxe ions, being actually such to the quantities from bence :=zxx ziy x log yt (by substituting yt which they are immediately derived; and thereiore, they may be found, in the same manner, by the for 2) y'<x log. y + xy. general rules already delivered. is the fluxion of Thus, by the 4th rule, the first fluxion of *3 .is the proposed exponential y*; which therefore consists of two terms, of which the one is the 3x**; and if x be supposed constant, or if the fluxion of the proposed quantity considering the root x be generated with an equable celerity, the esponent only as constant, and the other is the fluxion of 3x2x, or 3xxi?, will be 3x x 211=6xx", fuxion of the same quantity considering the root which is the second Auxion of *3; and 613 will be as constant. its third fluxion : but if the celerity with which * IX. 1o find the plurion of a rectangle, when one side x increases, and the other y decreases.-- In this is generated be variable, either increasing or decase the fluxion of the decreasing quantity is nega- creasing, then å being variable, will have its fluxtire with respect to that of the increasing quan- ion denoted by ä, &c. In this case the fluxion of tiy (see the beginning of this article), and therefore, the sign of the term affected with it ought to 3.1" x x will be, by the 2d and 4th rules, 6xixit be charged; e. gr. the fluxion of the rectangle 3x2xä=6012 +3127, the second fluxion of 73. *y in these circumstances will be expressed by And the third fluxion of x3 obtained in like manay-zy. ner from the last, will be 6x x x2 + 6x * 2x + X. To find the Aurions of sines, cosines, &c.Suppose we require the fluxion of sin z, that is, 618x8+ 3x27=633 + 187 ir + 3x2 č. Thus also, if the sine of the angle or arc denoted by z, we must suppose that by a motion of one of the legs in y=ni"-12 then y=nx n–1xs cluding the angle, it becomes z+z, then sin (z+z) and if 23 = x y, then 22% = xy + - sin z is the fluxion of sin z. But according to the formulæ for the sines of sums of arcs (see y r, &c. If the function proposed were ax, we should SinE and TRIGONOMETRY), we have sin (z+z)= find (a1n). i: the factors na and å being ein z cos 2+ sin 2 cos = the radius being assumed equal to unity. But the sine of an arc indefinitely regarded as constant in the first fluxion nar smail does not differ sensibly from that arc itself, Rom its cosine differ perceptibly from radius: hence , to obtain the second Auxion it will suffice to si sin l. make x ax (ax?) snar (axh): 1x33; 5 In like manner, the fluent flow and to multiply the result by of 4axx is 241?; nac. But (* )•=(1-1) * I we have, of 373 is 2x]; therefore, n+1 (ar")=n(n-1) ar of aing arn In a similar manner, we find, uti =n(n-1)(1-2) of (ax") * n(n-1)(n-2)(n-3) ax" 14, &c. And the fluxional coefficients will have the fol. of (a3 + 23)477 is st (43 +23)5. lowing values : III. When the root under a vinculum is a compound quantity; and the index of the part or factor with. (ar"); oul the vinculum increased by 1, is some multiple ef that under the vinculum.—Put a single variable (axn). letter for the compound root; and substitute its =n(n-1) ai powers and Auxion 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 =n(n-1)(n-2) ai can then be applied. So, if the given fuxion be ř=(a? +) 193 (axn) :: -=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 ** within the saine; Hence it is manifest that in the case where the therefore putting e=a? +*2, thence x:=s—ao, and exponent n is a positive whole number, the func- its Auxion is 2xx=2; hence then x3g = x3= tion aru has only a limited number of fluxions, of which the most elevated is (•) n ax"=n(n-1) $5 (z–a?), and the given quantity F or (1–2)... 2.1.axr; an expression which is nó (a2 + x2) 1x3de is = 1 :(z-a?) or = longer susceptible of fluxing, since it contains no more variable quantities. fa2d1c; and the fluent of each term gives Inverse method of fusions, relates to the finding Funden:42:= 3:1 (tós-ja); or, by substitutof fluents, and is of great utility though of con. siderable difficulty ing the value of : instead of it, the same fluent is As it is orly in certain particular forms and 3(a2 + x2) ix (x2-x.(2), or yća? + ** }.?-?u?. cases that the fluents of given fluxions can be found; there being no method of performing this IV. When there are several terms involving two er universally a priori, by a direct investigation; multiplied by the other quantity or quantities... more variable quantities, having the fiuxion of each like finding the fluxion of a given fluent quantity; Take the Auent of each term, as if there was only we can do little more than lay down a few rules for such forms of Auxions as are known, from the one variable quantity in it, namely that whose direct method, to belong to such and such kinds Auxion is contained in it, supposing all the others of fuents or flowing quantities; and these rules, to be constant in that term; then if the fluents of it is evident, must chiefly consist in performing that quantity will be the fluent of the whole. all the terms so found be the very same quantity, such operacions as are the reverse of those by which the fluxions are found to given flowing Thus, if the given Auxion be xy + xy. Then, quantities. The principal cases of which are as the fluent of xy is xy, supposing y constant; and follow: the fluent of zy is also xy, when m is constant ; 1. To find the fluent of a simple fuxion ; or that therefore the common resulting quantity ky is the in which there is no variable quantity, and only one fluxional quantity:- This is done by barely substi- required fluent of the given fluxion xy + xij, And in like manner the fluent of tuting the variable or flowing quantity instead of its fluxion, and is the result or reverse of the no- xyz + xyz +ryc is xyz. tation only. Thus, v. when the given fuxional expression is in this The fluent of ax is 2x. The fluent of ay + 2y is ay + 2y. viz, a fraction including two quanti ya The fluent of va? +1? is va? +*?. ties, being the furion of the former drawn into the II. When any power of a flowing quantily is latter, minus the flusion of the latter drawn into the multiplied ly the fluxion of the root.—Then, having former, and divided by the square of the latter: substituted, as before, the flowing quantity for its then the fluent is the fraction or of the fuxion, divide the result by the new index of the power. Or, which is the same thing, take out, or former quantity divided by the latter. That is, divide by, the fluxion of the root; add 1 to the **-*y index of the power; and divide by the index so the fluent of is increased. у ys So if the Auxion proposed be 315%; 2**y* -x?yy and the fluent of Strike out i then it is ya add 1 to the index, and it is divide by the index 6, and it is 2x6 or 116; Though the examples of this case may be per formed by the foregoing one. Thus the gives which is the fluent of the proposed fuxion 3x*x. Auxion form ty — x1 y 3.15; 3x6; y-x* reduces to orxiu; of * or nxnzG; the fluxion of this is G=nax ; therefore Ġ :Ė :: G:f be *Y or comes n?"-1: is + xy -1 or -1. 1-* & + x3 which the fluent of is - when y is constant; fluent sought. y For a 21 ex. suppose it be proposed to find the and the fluent of xyy when Auent of xy + xy. Here F= y + xy; then, writing y * for å, and y for y, it is xy + xy or 2xy=G; the x is constant; and therefore, by that case, is fluxion of which is 2xy + 2xy=G; then Ġ : 1 :: G:F becomes 2xy + 2xy: xy + xy :: 2xy : ay=F, ху — ху the floent of the whole the fluent sought. ya VIII. But fluents are more generally found by VI. When the furion of a quantity is divided by means of a table of forms of fluxions and fuents, the quantity itself. Then the Auent is equal to the in which comparing any proposed Bluxion with a byperbolic logarithm of that quantity; or, which corresponding form given in the table, the fluent is the same thing, the Auent is equal to 2-30258509, of it will be found. A very useful table of this &c. multiplied by the common log. of the same kind is inserted under the article Fluent in Dr. quantity. Hutton's Math. Dictionary, and illustrated by Thus the fluent 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 of *, is the hyp. log. of x; amply shewn by Holliday and succeeding writers. IX. To find fluents by means of infinile series. When a finite form cannot be found to agree with of ;is ? x hyp. log. of x, or hyp. log. of **; a proposed fiuxion, it is then usual to throw it into an infinite series, either by division, or ex traction of roots, or by the binomial theorem, &c. of is the hyp. log. of a+x; at after which the fuenis of all the terms are taken separately. of , is the hyp. log. of a+*3; For ex. To find the fluent of 1+*-* Here, by dividing the numerator by the denomi. of is hyp. log. of (x + x2 +42); nator, this becomes * – 2xx+ 3x?*— 5*3*+ 8.143, &c.; and the fluents of all the terms being taken, give of is hyp.log: (x+4+V *?£ lax); sought. *— 2? + 13- ***+-, &c., for the fluent V x lax Some excellent examples of this method may be seen in Simpson's Fluxions, vol. i. hyp. log.of 4+* To correct a Fluent.-The fluent of a given sur, ion, found as above, sometimes wants a correction, 208 Vai to make it contemporary with that required by of is hyp. log. of the problem under consideration, &c.: for the *18 at va? £*? fluent of any given fuxion, as x, may lie either x VII. Many Anenis may be found by the direct (which is found by the rule) or it may be 1 Er, authod of fiunions, thus. —Take the fuxion again that is x plus or minus some constant quantity of the given fluzional expression, or the 2d Auxion because both x and x Ec have the same fluxion :: of the fluent sought; into which substitute and the finding of the constant quantity, is called correcting the fluent. Now this correction is to ya be determined from the nature of the problem in for i and for y, &c. that is, make x, x, z, as hand, by which we come to know the relation which the fluent quantities have to each other at also y, y, y, &c. in continual proportion. Then some certain point or time. Reduce therefore divide the square of the given fusional expression the generai fluential equation, found by the rules by the 2d furion, just found, and the quotient will above, to that point or time; then if the equation be the fluent sought in many cases. be true at that point, it is correct; but if not, ic Or the same rule may be delivered thus. In wants a correction, and the quantity of that corthe given Aurion ! write : for Å, y for y, &c. rection is the difference between the two general and call the result G, taking also the Auxion ticular state. Hence the general rule for the cor sides of the equation when reduced to that parof this quantity, ċ; then make Ġ:8::G:F, rection is this: so shall the fourth proportional F be the fluent, Connect the constant, but indeterminate, quan. as before. This is the rule of M. Paccassi. tity c with one side of the Nuential equation, as It may be proved whether this be the true determined by the foregoing rules; then, in this fuent, by taking the fluxion of it again, which, equation, substitute for the variable quantities if it agree with the proposed Auxion, 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 particular state of the equation find the value of Thus, if it be proposed to find the fluent of c, the constant quantity of the correction. 21*-';. Here F = 147-1; write first * for x, Ex. To find the correct fluent of 2 = axix. First Das a a-X a fails. the general Auent of this is z=214, or z=0xt+c, the preceding problem, the abscissa AP had beers taking in the correction c. supposed equal to x, while we only required the Now if it be known that z and r begin together, magnitude of the segment CBPQ. For we should or that z=0, when ==0; then writing o both for find the fluxion of this area =ivpr, and its fluent # and z, the general equation becomes ()=0 +c, or c=0; so that the value of c being 0, the correct = **Vpx. But this expression is that of the total fluents are z=a14. area APQ, and the area demanded should vanish But if z bo =0, when r is =b, any known quan- and the preceding Auent becomes reduced to when AP=AB. We must therefore suppose I=i, tity; then substituting 0 for a, and b for 1, in the appa, which being deducted from the former exwhich is found o=-ab'; and this being written pression, will give for the true value of the area for it in the general equation, this becomes z= sought, fr pr-a v pa. 014- abt, for the correct or contemporary fluents. A sketch of the principles of fluxions being now delivered, we may next say a little respectOr lastly, if it be known that e is = some quan. ing the chief writings and improvements that have tity d, when « is equal some other quantity, as b; been made by divers authors, since the first disthen substituting d for z, and b for. 1, in the gene- covery of them : indeed some of the chief improveral fluential equation x=214 +c, it becomes de @b9 +; and hence is deduced the value of the cor ments may be learned by consulting the preface rection, viz. c-d-ab4; consequently, writing to Dr. Waring's Meditaticnes Analyticæ. this value for cin the general equation, it becomes Auxions to a considerable degree of perfection; as The inventor himself brought the doctrine of z=ant - abt + d, for the correct equation of the fluents in this case. may be seen by many specimens of this science, And hence arises another easy and general way his Tract on Quadratures, and in his Treatise on given by him ; particularly in his Principia, in of correcting the fluents, which is this: In the Fluxions, published by Mr. Colson; from all of general equation of the fluents, write the particu- which it will appear, that he not only laid down lar values of the quantities which they are known to have at any certain time; then subtract the the whole theory of this method, both direct and inverse; but also applied it in practice to the sosides of the resulting particular equation, from lution of many of the most useful and important the corresponding sides of the general one, and the remainders will give the correct equation of problems in mathematics and philosophy. the fluents sought. So, as above, Various improvements however have been made by many illustrious authors on this science; parthe general equation being, a=234; write d for , and b for then d=alt; ticularly by John Bernoulli, who treated of the fluents belonging to the fluxions of exponential hence by subtraction z-d=axt-ab, expressions ; James Bernoulli, Craig, Cheyde, or z=ext-ab4 - d, the correct fluents as before. (Hutton's Dict.) Cotes, Manfredi, Riccati, Taylor, Fagnanus, ClaiAs this is a very important part of the doctrine Le Grange, Emerson, Simpson, Landen, Waring, raut, D'Alembert, Euler, Condorcet, Walmesley, of fuxions, it may be as well to illustrate it a little Bezout, Bossut, Lacroix, &c. There are several farther. Let us demand, therefore, the area of a parabola, commencing not at the summit A, but other treatises also on the principles of Fluxions at the point B (fig. 11. Pl. 68.), taken upon the Rowe, Vince, &c. &c., delivering the elemcnis of by Hayes, Newyentyt, L Hôpital, Hodgson, axis at any distance whatever from that summit. this science in an easy and familiar manner. Here, then, the abscissa BP being represented by The elements of the doctrine of fluxions have a', the distance AB from the vertex by d, and the been delivered by its great author in so concise a parameter by p, we have for the equation of the manner, as to give occasion to the most ingenious curve, y=1' pa+px=PQ; and the sluxion of the bishop Berkeley to represent it as founded on inarea will be gå=xiv pa+px: the fluent of this This author, in a letter under the title of the conceivable principles and full of false reasonings. found by the usual methods will be Analyst, published in the year 1734, has been at great pains to convince his readers, that the or jects, principles, and inferences of the modern 个 p analysis by fluxions, are not more distinctly con. 1 AP · PQ. Now, it is evident that the area re- ceived, or more evidently deduced, than religious quired should be nothing, when r or BP becomes mysteries and points of faith. He says he does nothing. The area above is therefore too great not controvert the truth of the conclusions, but by the area ABC; and this area ABC is found by only the logic and method of mathematicians. making x=() in the above expression; for it then An answer to the objections of the Analyst apbecomes a v pa={AB:BC. Whence we see that peared very early under the name of Philalethes to apply this correction to the fluent, it is requi- Cantabrigiensis, since known to be Dr. Jurin. site to know what it becomes when x is made And various others were published on the same equal to nothing; and if the result be positive, it side of the question. But ihe most able defences must be subtracted; but if it be negative it must of the doctrine of fluxions were made by Mr. be added : for in the former case the fluent is too Robins (see his Tracts, vol. ii.) and by Mr. Macgreat, and in the latter it is too little. laurin, in his very elaborate and excellent Trea. There are cases also where by the conditions of tise on Fluxions. The demonstrations of these the problem, we know that the fluent ought to be two gentlemen are, we think, sufficient to satisfy equal to nothing when the abscissa or r, instead the most scrupulous: to them, therefore, we with of being 0, is of a determinate magnitude as a. It pleasure refer. Mr. Woodhouse, who, in his is then requisite to suppose r in the fluent equal Principles of Analytical Calculation, has revived to a; and if the result is positive to subtract it many of Berkeley's objections with greater mathefrom the fluent, since it is too great; and, on the matical talent, but with less meiaphysical acutecontrary, to add it if the result is negative, since ness, says, the prolixity of the reasonings of Macit is too little. This would have happened if, in laurin and Robins "confirm the notion that the 2 pa + prw_pa+p+=}(a+x)V pa + px method they defend is an incommodious one." off become worms, and thence flies, like the This remark we think uncandid: the prolixity is British caterpillar. Such was the account orioccasioned not by want of simplicity in the me. ginally given of this extraordinary production. thod they defend, but by the tortuous intricacy and But several boxes of these fies having been sent obscurity of many of the objections, which requir- to Dr. Hill for examination, his report was ed much time to be pursued throughout, and, as this : “ There is in Martinique a fungus of the several of them were subtile and specious, much labour and room to refute them completely. clavaria kind, different in species from those With regard to the comparative perspicuity of the hitherto known. It produces soboles from its two methods of establishing the principles, from sides; I call it, therefore, clavaria sobolifera. the doctrine of motion and from pure analysis, we It grows on putrid animal bodies, as our funcannot hesitate to say that the former has the gus ex pede equino from the dead horse's hoof. preference: and we are convinced that every un. The cicada is common in Martinique, and in biassed reader of the treatises of Maclaurin and its nympha state, in which the old authors Simpson on the one hand, and of Bossut and La- call it teitigometra : it buries itself under dead croii on the other, will agree with us in opinion. leaves to wait its change; and, when the seaYet, we are constrained to acknowledge, notwithstanding this, that the principal improvements of the clavaria find a proper bed in this dead son is unfavourable, many perish. The seeds and extensions of the modern analysis, during the last forty years, have been made by continental, the cicadæ in the British museum; the clava insect, and grow The ietiigometra is among sot by British, mathematicians. For the various applications of fluxions to ria is just now known. This is the fact, and practical purposes, see the articles AsymPTOTE, all the fact; though the untaught inhabitants ISILLION, MAXIMA, TANGENT, &c. suppose a fly to vegetate, and though there is a T. FLY. v. n. pret. flew or fled; part. fled Spanish drawing of the plants growing into a or floxe: fled is properly from flee. (fleogan.) trifoliate tree, and it has been figured with the 1. To more through the air with wings (Shak. creature flying with this tree upon its back.” speare). 2. To pass through the air (Jol). 3. Edwards has taken notice of this extraordinary To pass away (Prior).. 4. To pass swiftly production in his Gleanings of Natural Hisrtoy. (Pope). 5. To move with rapidity (Dryden). Fly, in mechanics, is a name given to a 6. To part with violence (Shakspeare). 7. certain appendage to many machines, either as a "To break; to shiver; to burst asunder with a regulator of their motions, or as a collector of power. sodden explosion (Suifi). 8. To run away ; In the first case the fly is a heavy disk or hoop, to flee (Prior). 9. To Fly at. To spring or other mass of matter balanced on its axis, and with violence upon; to fall on suddenly so connected with the machinery as to turn brisk(South). 10. To Fly in the face. To insult ly round with it. This may be done with the (Stijn). 11. To Fly in the face. To act in view of rendering the motion of the whole more dehance. 12. To Fly off. To revolt (Addi- of the accelerating forces, or of the resistances regular, notwithstanding unavoidable inequalities ). 13. To Fly out. To burst into passion occasioned by the work. It becomes a REGULA(Ben Jonson). 14. To Fly out. To break TOR. Suppose the resistance extremely unequal, out into licence. 15. To Fly out. To start and the impelling power perfectly constant; as violently from any direction (Bentley). 10. when a bucket wheel is employed to work one To let Fly. To discharge (Glanville). pump. When the piston has ended its working TO FLY. v. . 1. To shun; to avoid ; to stroke, and while it is going down the barrel, the skeline (Shakspeare). 2. To refuse associa- power of the wheel being scarcely opposed, it action with (Dryden). 3. To quit by Aight celerates the whole machine, and the piston ar(Dryden). 4.° To attack by a bird of prey able velocity. But in the rising again, the wheel rives at the bottom of the barrel with a consider(Bacon). Fly, in eatomology. See MUSCA. is opposed by the column of water now pressing Fix (Honeysuckle), in botany. See Loxi- wheel; and when the piston has reached the top of on the piston. This immediately retards the the barrel, all the acceleration is undone, and is to Flv (lloneysuckle), African. See Hale begin again. The motion of such a machine is LERIA, very hobbling: but the superplus of accelerating FLY-CATCHER, in ornithology. See Mus- force at the beginning of a returning stroke will CICAPS, not make such a change in the motion of the maFLY (Spanish). See CANTHARIDES. chine if we connect the fly with it. For the ac. FLY-TRAP (Venus's), See DioNÆA. celerating momentum is a determinate quantity. FLY THE HEELs, in the manage, is when Therefore, if the radius of the fly be great, this a horse obeys the spurs. momentum will be attained by communicating a Fly (Vegetable), a very curious natural small angular motion to the machine. The moproduction, chiefly found in the West In- mentum of the fly is as the square of its radius; dies, and thus improperly denominated. Ex- therefore it resists acceleration in this proportion; cepting that it has no wings, it resembles and although the overplus of power generates the same momentum of rotation in the whole ma. the drone both in size and colour more than chine as before, it makes but a small addition to any other British insect. In the morth of its velocity. If the diameter of the fly be doubled, May it buries itself in the earth, and begins to the augmentation of rotation will be reduced to regetate. By the latter end of July, the tree is one-fourth. Thus, by giving a rapid motion to a arrived at its fuld growth, and resembles a co- small quantity of matter, the great acceleration, sal branch ; and is about three inches high, during the returning stroke of the piston, is pre2nd bears several little pods, which dropping repted. This acceleration continues, however, YOL. V. |