460

FLUXIONS.

titiesare found, as of y which put equal to z, then by the proper-ties of logarithms the log. z = x x log. y, hence by taking the

* y

fluxions — = w X log. y - fix — (art. 25, and 19,) therefore *z y

z *y x x

=s z x x log. y -j-= (by puling y for its equal z) y x X

y

log. y r y y = the fluxion of the exponential y ;which consists of two terms, and it is evident that if either of thequantities r or y be constant, the term into which its fluxion enterswill vanish, the fluxion being equal to nothing: hence when the

root?/ is constant, the fluxion of the exponential quantity y x isequal y X x x log. y. If jr be constant it ceases to be an exponen-tial, and the fluxion is x y x ~ l y as in articles 22, 23, and 24.

(28.) The second fluxion is derived from the first, and the thirdfrom the second &c. in the same manner as the first from the flowingquantity. Tims the fluxion of x 3 is 3 x 2 £ ; (art. 22,) its secondfluxion is6x i 2 -j-3x 2 x, (art. 19, and 22,) &c. If x be con-stant, x = 0, and the second fluxion of x 3 is 6 x £ 2

(29.) In the foregoing articles, it has been supposed that thefluents increase, and that their fluxions are therefore positive ; butwhen it happens that the fluents decrease, then the fluxions arenegative and must be written with a contrary sign to that deter-mined by those articles; thus if in the product x y, y decreasewhile x increases, the fluxion of y is — y and the fluxion of the pro-duct is yx — xy.

(30.) From what has been advanced we may easily collect thefollowing practical rules for finding the fluxions of quantities.

1st. For the fluxion of any pozver or root of a flowing quantity.Multiply by the index and by the fluxion of the root, and diminishthe index by unity, (art. 22, 23, 24.)

2nd. For the fluxion of a product. Multiply the fluxion ofeach quantity by the product of all the rest, and connect them bytheir proper signs, (art. 19, 20.)

3rd. For the fluxion of a fraction. From the fluxion of thenumerator multiplied into the denominator, subtract the fluxion ofthe denominator multiplied into the numerator, and divide by thesquare of the denominator, (art. 21.)

4th. For the fluxion of a logarithm. Divide the fluxion of thequaritity by the quautity itself and multiply the result into the mo-dulus of the system of logarithms, (art. 25. 26.)

5th.,For the fluxion oj an exponential quantity. Multiply thegiven quantity into the fluxion of its exponent and into the hyp.log. of the root, and add the fluxion of the given quantity, foundby the first rule, as if the root only were variable. The last partvanishes, and therefore need not be found when the root is constant,(art. 27.)

6th. In complex cases the preceding rules are to be usedjointly. Or the following general rule will be found useful w henthere are several variable quantities. Take the fluxion of thegiven quantity as often as there are. variable letters or symbols in it,supposing first only one of them variable, and the rest constant;then another variable, and the rest constant; and so on till each hasbeen considered as variable, and connect all these fluxions by theirproper signs.

Note. When the given quantity consists of several terms, findthe fluxion of each term separately, and connect them all togetherby their proper signs. The following example is added .to illus-trate the rules, particularly the sixth.

2 a x 3 y-

Examp. To find the fluxion of-1- hyp. log, is —

z

cy z X ’ where x, y and z are flowing quantities. Proceeding as di-rected in the first part of rule 6, the fluxion wdl stand thus;

lax 3 yf -j- 6 a y' 3 a 4 x x z— 2a x 3 if £ x £ -(- z x

-:---:- +- cz x y —

z~ xz

^z r x x hyp.log. s + xz x ~ 'a ) X cy. Or, proceeding accord-ing to the latter part of the 6th rule it will stand as follows, viz.

4 ax 3 yy 6 ay*x*x- 1 -

2 ax 3 y 1 £ £ x

x . x .

, , — cz y — cyz ,x

Z X

X hyp. log. z — cxyz x ~ 1 a, which is the same as the foregoing.The second, third, &c. fluxions are found by the same rules (art. 28.)

Thus the fluxion of a 3 is 3x 3 £ ; its second, 6x£ 2 -f-3x 2 .V ;• and soon : but it x be invariable, x = 0, and the second fluxion -of x 3 =6x.v 2 .

Prob. I. To determine Maxima and Minima.

(31.) When a quantity increases, its fluxion is positive ; whenit decreases, it is negative ; therefore when it is just betwixt in-creasing and decreasing, its fluxion is = 0.

Hale. Find the fluxion, make it = 0, whence an equation willresult giving an answer to the question.

(32.) Examp. 1. To determine the dimensions of a cylindricmeasure ABCD, fig. 9, open at the top, w hich shall contain agiven quantity under the least internal superficies possible.

Let the diameter All = x, the altitude AC = y, the givenquantity, or content = c, and £ of the circumference of a circlewhose diameter is unity = p =,78539, &c. Then p x 3 y = c, and4c

4 pxy — —, hut Apxy = the concave superficies, thereforex

4 c

-hP** = the whole internal superficies, which is to be a mi-

x

4 c -v

nimum, therefore, its fluxion-(- 2 p x x = 0, hence px 3

3 x 2

= 2 c, and x

Since p x 3 = 2 c, and p x l y = e, p x 3

= 2 p x s //, and x = 2 y ; whence y is also known, and the dia-meter of the base must be just double the altitude.

Examp. 2. Let ABC represent a cone, AC the diameter of thebase ; to inscribe in it the greatest cylinder DFGI, fig. 16.

. ~ 11= i

Put p =,78539, &c., then since A C == a . BIT=i . BEp a 2 x 2

= x,-= the area of the end D E F of the cylinder; hence,

b 3

p a 2 x 2 _

the content of the cylinder =- x b — x = max. orx 2 X

_ b 2

b—x = b x* — x 3 = max. 2bxx — 3 x 2 x — 0 ; hence, x —2 1

— b ; therefore E II = — b.

3 3

F.xamp. 3. Given x + y + z = a, and x y* z 3 a maximum, tofind x, y, and z.

As x, y, z, must have some certain determinate values to an-swer these conditions, let us suppose such a value of ;/ to remainconstant, whilst x and z vary till they answer the conditions, andthen x -J- £ = 0, and z 3 x -f- 3 xz 2 * = t); hbnee, x = — £ = —3 x z 2 £ 3 X £

-=-, .: z = 3 r. Now let us suppose the value of

z 3 z

z to remain constant, and x and y to vary, so as to satisfy the con-ditions ; then A -f- y = 0, y* x 4- 2xi/y = 0 ; hence, x =—y =2 x y y 2 xy

---, y — 2 x ; substitute in the given equa-

V

tion these values of y and z in terms of x, and x-j-2x-j- 3x = rt, ° r1 1 1

6 x = a ; hence, x =— a ; :.y— — a ; z = — a. In like man-C 3 2

tier, whatever be the number of unknown quantities, make antone of them variable with each of the rest, and the values of eachin terms of that one quantity will he obtained ; and by substiUi'ling the values of each in terms of that one, in the given equa*tion, you will get the value of that quantity, and hence the value*oi the others.

(33.) Fig. 10. To find the longest and shortest ordinates of anycurve, DEF, whose equation is known,

4 Make