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EXAM. 10. To find the fluent of (+/-) x3y2.
EXAM. 11. To find the fluent of (+
EXAM. 12. To find the fluent of
EXAM. 13. To find the fluent of
EXAM. 14. To find the fluent of
EXAM. 15. To find the fluent of
EXAM. 16. To find the fluent of
EXAM. 17. To find the fluent of
EXAM. 18. To find the fluent of
EXAM. 19. To find the fluent of
EXAM. 20. To find the fluent of
EXAM. 21. To find the fluent of
EXAM. 22. To find the fluent of
EXAM. 23. To find the fluent of
EXAM. 24. To find the fluent of
EXAM. 25. To find the fluent of
EXAM. 26. To find the fluent of
√ ax2 + x/2/2
EXAM. 28. To find the fluent of 2x√2x
EXAM. 30. To find the fluent of 3axx.
EXAM. 31. To find the fluent of 32x log. z + 3xxTM ̃ż.
EXAM. 33. To find the fluent of (2 + x) x2x.
To find Fluents by Infinite Series.
44. When a given fluxion, whose fluent is required, is so complex, that it cannot be made to agree with any of the forms in the foregoing table of cases, nor made out from the general rules before given; recourse may then be had to the method of infinite series; which is thus performed:
Expand the radical or fraction, in the given fluxion, into an infinite series of simple terms, by the methods given for that purpose in books of algebra; viz. either by division or extraction of roots, or by the binomial theorem, &c; and multiply every term by the fluxional letter, and by such simple variable factor as the given fluxional expression may contain. Then take the fluent of each term separately, by the foregoing rules, connecting them all together by their proper signs; and the series will be the fluent sought, after it is multiplied by any constant factor or co-efficient which may be contained in the given fluxional expression.
45. It is to be noted however, that the quantities must be so arranged, as that the series produced may be a converging one, rather than diverging: and this is effected by placing the greater terms foremost in the given fluxion. When these are known or constant quantities, the infinite series will be an ascending one; that is, the powers of the variable quantity will ascend or increase; but if the variable quantity be set foremost, the infinite series produced will be a descending one, or the powers of that quantity will de- ́ crease always more and more in the succeeding terms, or increase in the denominators of them, which is the same thing.
For example, to find the fluent of
1 + x
Here, by dividing the numerator by the denominator, the proposed fluxion becomes-2xx+3x2x − 5x3x+8x1x−&c; then the fluents of all the terms being taken, give
&c, for the fluent sought.
Again, to find the fluent of x1 − x2.
Here, by extracting the root, or expanding the radical quantity 1 - x2, the given fluxion becomes
&c. Then the fluents of all
the terms, being taken, give x - Zx3 for the fluent sought.
EXAM. 7. Given 2 = 3*a + x, to find z.
EXAM. 13. Given &= 2 x3x3 + x2 + x3, to find
EXAM. 14. Given 2 = 5*√√ax
xx, to find z.
To Correct the Fluent of any Given Fluxion.
46. The fluxion found from a given fluent, is always perfect and complete; but the fluent found from a given fluxion is not always so; as it often wants a correction, to make it contemporaneous with that required by the problem under consideration, &c: for, the fluent of any given fluxion, as x may be either x, which is found by the rule, or it may berc, or x - c, that is a plus or minus some constant quantity c; because both r and have the same fluxion, and the finding of the constant quantity c, to be added or subtracted with the fluent as found by the foregoing rules, 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, supposed to be found by the foregoing rules, to that point or time; then if the equation be true, it is correct; but if not, it wants a correction; and the quantity of the correction, is the difference between the two general sides of the equation when reduced to that particular point. Hence the general rule for the correction is this:
Connect the constant, but indeterminate, quantity c, with one side of the fluential equation, as determined by the foregoing rules; then, in this equation, substitute for the variable quantities, such values as they are known to have at any particular state, place, or time; and then, from that particular state of the equation, find the value of c, the constant quantity of the correction.
47. EXAM. 1. To find the correct fluent of ż = ax3x. The general fluent is z = ax1, or z = ax+ + c, taking in the correction c.
Now, if it be known that z and begin together, or that z is 0, when x = 0; then writing 0 for both x and x, the general equation becomes 0 = 0 + c, or = c; so that, the value of c being 0, the correct fluents are z = = ax4.
But if z be 0, when a is b, any known quantity; then substituting 0 for z, and b for x, in the general equation, it becomes 0= ab+c, and hence we find c = which being written for c in the general fluential equation, it becomes z = axa — aba, for the correct fluents.
Or, if it be known that z is some quantity d, when is some other quantity as b; then substituting d for z, and b for x, in the general fluential equation z = ar+ + c, it becomes dab+c; and hence is deduced the value of the correction, namely, c = d - ab+; consequently, writing this value for c in the general equation, it becomes
z = axt
ab+d, for the correct equation of the fluents
48. 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 or position; 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, the general equation being z = ax; write d for z, and b for æ, then d= hence, by subtraction,
or z = ax1
Z - d = ax2 — ab1,
EXAM. 2. To find the correct fluents of
=0 when x is = a.
= 5xx; z being
3x √ a + xi
EXAM. 3. To find the correct fluents of % = z and r being = 0 at the same time.
EXAM. 4. To find the correct fluent of ✯ =
a + x;
posing z and to begin to flow together, or to be each O at the same time.
EXAM. 5. To find the correct fluents of =
posing z and a to begin together.
a2 + x2
OF MAXIMA AND MINIMA; OR, THE GREATEST AND LEAST MAGNITUDE OF VARIABLE OR FLOWING QUANTITIES.
49. MAXIMUM, denotes the greatest state or quantity attainable in any given case, or the greatest value of a variable quantity: by which it stands opposed to Minimum, which is the least possible quantity in any case.