!

OF SECOND, THIRD, &c. FLUXIONS.

315

Also, the higher orders of fluxions are found in the same manner as the lower ones. Thus,

the first fluxion of y3 is

3y2y;

its 2d flux. or the flux. of 3y2y, con

sidered as the rectangle of 3y2, ) 3y2ÿ+6yy2
and y, is

and the flux. of this again, or the 3d

flux. of y3, is

3

3y2 ÿ +18y ÿÿ + 6y3,

29. In the foregoing articles, it has been supposed that the
fluents increase, or that their fluxions are positive; but it
often happens that some fluents decrease, and that therefore
their fluxions are negative: and whenever this is the case,
the sign of the fluxion must be changed, or made contrary to
that of the fluent. So, of the rectangle xy, when both ≈ and
y increase together, the fluxion is xy + xy, but if one of
them, as y, decrease, while the other, a, increases; then, the
fluxion of y being — y, the fluxion of xy will in that case be
xy-xy. This may be illustrated by
the annexed rectangle, APQR = XY,
supposed to be generated by the mo-
tion of the line PQ from A towards c,
and by the motion of the line RQ from
B в towards a: For, by the motion of
re, from a towards c, the rectangle
is increased, and its fluxion is + xy;
but, by the motion of RQ, from в to-
wards A, the rectangle is decreased, p
and the fluxion of the decrease is xy; therefore, taking the
fluxion of the decrease from that of the increase, the fluxion
of the rectangle xy, when x increases and y decreases, is xy

Je

A

P C

xy.

REMARK BY THE EDITOR.

.

The fluxion of the algebraic quantity xy is properly yx+xy in all cases of increase or decrease. We should always use the signs of the fluxions of algebraic expressions as those signs arise from the known rules, without considering whether the quantities increase or decrease; but in denoting, algebrai cally, the simple fluxions of geometrical quantities, we should prefix the sign minus to the fluxions of such as decrease: and thus we may, in any case, use the fluxions of algebraic equations, together with the fluxions derived from geometrical figures, without embarrasment or apprehension of error.

30. We

RULES FOR FINDING

30. We may now collect all the rules together, which have - been demonstrated in the foregoing articles, for finding the fluxions of all sorts of quantities. And hence,

7.

316

1st, For the Auxion of any Power of a flowing quantity. Multiply all together the exponent of the power. the fluxion of the root, and the power next less by 1 of the same root.

2d, For the fluxion of the Rectangle of two quantities.-Multiply each quantity by the fluxion of the other, and connect the two products together by their proper signs.

3d, For the fluxion of the Continual product of any number of flowing quantities.-Multiply the fluxion of each quantity by the product of all the other quantities, and connect all the products together by their proper signs.

4th, For the fluxion of a Fraction. From the fluxion of the numerator drawn into the denominator, subtract the fluxion of the denominator drawn into the numerator, and divide the result by the square of the denominator.

5th, Or, the 2d, 3d, and 4th cases may be all included under one, and performed thus.-Take the fluxion of the given expression as often as there are variable quantities in it, supposing first only one of them variable, and the rest constant; then another variable, and the rest constant, and so on, till they have all in their turns been singly supposed variable, and connect all these fluxions together with their own signs.

6th, For the fluxion of a Logarithm.-Divide the fluxion of the quantity by the quantity itself, and multiply the result by the modulus of the system of logarithms.

Note. The modulus of the hyperbolic logarithms is 1, and the modulus of the common logs. is

0.43429448.

•

8th, For the fluxion of an Exponential quantity having the Root Variable-To the fluxion of the given quantity, found by the 1st rule, as if the root only were variable, and the fluxion of the same quantity found by the 7th rule as if the exponent only were variable; and the sum will be the fluxión for both of them variable.

7th, For the Auxion of an Exponential quantity having the Root Constant.-Multiply altogether, the given quantity the fluxion of its exponent, and the hyp. log. of the root.

Note. When the given quantity consists of several terms, find the fluxion of each term separately, and connect them all together with their proper signs.

31. PRACTICAL

32. IT has been observed, that a Fluent, or Flowing Quantity, is the variable quantity which is considered as increasing or decreasing. Or, the fluent of a given fluxion, is such a quantity, that its fluxion, found according to the foregoing rules, shall be the same as the fluxion given or proposed.

·

33. It may further be observed, that Contemporary Fluents, or Contemporary Fluxions, are such as flow together, or for the same time. When contemporary fluents are always equal, or in any constant ratio: then also are their fluxions respectively either equal, or in that same constant ratio. That is, if x y, then is x = y; or if ry:n : 1, then is x y :: n 1; or if any, then is

·

ny

34. It is easy to find the fluxions to all the given forms of fluents; but, on the contrary, it is difficult to find the fluents of many given fluxions; and indeed there are numberless

cases