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a + x
2, gives a = 3x; conseq.
, and =, which is the constant factor for every succeeding term; also,2m 2 × 43429448190868588964; therefore the calculation will be conveniently made, by first dividing this number by 3, then the quotients successively by 9, and lastly these quotients in order by the respective numbers 1, 3, 5, 7, 9, &c, and after that, adding all the terms together, as follows:
Sum of the terms gives log. 2=301029995
TO FIND THE POINTS OF INFLEXION,
70. THE Point of Inflexion in a curve, is that point of it which separates the concave from the convex part, lying between the two; or where the curve
changes from concave to convex, or from convex to concave, on the same side of the curve. Such as the point E in the annexed figures, where the former of the two is concave
towards the axis AD, from A to E, but convex from E to F; and on the contrary, the latter figure is convex from a to E, and concave from E to F.
71. From the nature of curvature, as has been remarked before at art. 28, it is evident, that when a curve is concave towards an axis, then the fluxion of the ordinate decreases, or is in a decreasing ratio, with regard to the fluxion of the absciss; but, on the contrary, that it increases, or is in an increasing ratio to the fluxion of the absciss, when the curve is convex towards the axis; and consequently those two fluxions are in a constant ratio at the point of inflexion, where the curve is neither convex nor concave; that is, is toy in a constant ratio, or or is a constant quantity. But constant quantities have no fluxion, or their fluxion is equal to nothing; so that in this case, the fluxion of
or of is equal to nothing. And hence we have this general rule:
72. Put the given equation of the curve into fluxions;
from which find either or. Then take the fluxion of
this ratio, or fraction, and put it equal to 0 or nothing; and
from this last equation find also the value of the same or Then put this latter value equal to the former, which will form an equation; from which, and the first given equation of the curve, x and y will be determined, being the absciss and ordinate answering to the point of inflexion in the curve, as required.
EXAM. 1. To find the point of inflexion in the curve whose equation is axa'y + x2y.
This equation in fluxions is 2axx = dj +2.2y + Tủ, * a2+x2
Then the fluxion of this quantity
made = 0, gives 2xx (ax —xy) = (a2 + x2) × (øx − xy−xj);
* a2 + x2
and this again gives
Lastly, this value of
being put equal the former, gives
and hence 2.r2 = q2 = x2,
∞ 3x2 = a2, and x = a, the absciss. Hence also, from the original equation,
a2 + x2 a2
a, the ordinate of the point of in
EXAM. 2. To find the point of inflexion in a curve defined by the equation ay = a/ax2 + xx.
EXAM. 3. To find the point of inflexion in a cùrve defined by the equation ay ar+x3.
EXAM. 4. To find the point of inflexion in the Conchoid of Nicomedes, which is generated or constructed in this manner: From a fixed point P, which is called the pole of the conchoid, draw any number of right lines PA, PB, PC, PE, &c, cutting the given line FD in the points F, G, H, I,
&c: then make the distances FA, GB, HC, IE, &c, equal to each other, and equal to a given line; then the curve line ABCE &c, will be the conehoid; a curve so called by its inventor Nicomedes.
TO FIND THE RADIUS OF CURVATURE OF CURVES.
73. THE Curvature of a Circle is constant, or the same in every point of it, and its radius is the radius of curvature. But the case is different in other curves, every one of which has its curvature continually varying, either increasing or decreasing, and every point having a degree of curvature peculiar to itself; and the radius of a circle which has the same curvature with the curve at any given point, is the radius of curvature at that point; which radius it is the business of this chapter to find.
74. Let AEe be any curve, concave towards its axis AD; draw an ordinate DE to the point E, where the curvature is to be found; and suppose EC perpendicular to the curve, and equal to the radius of curvature sought, or equal to the radius of a circle having the same curvature there, and with that radius describe the said équally
curved circle BEe; lastly, draw Ed parallel to AD, and de parallel and indefinitely near to DE: thereby making Ed the fluxion or increment of the absciss AD, also de the fluxion of the ordinate DE, and Ee that of the curve AE. Then put
x = AD, y = DE, 2 = AE, and r = CE the radius of curvature; then is Ed= x, de ≈ ÿ, and ɛe = ż.
Now, by sim. triangles, the three lines Ed, de, Ee,
are respectively as the three therefore
and the flux. of this eq. is GC.
or, because GC - BG, it is GC.
or x, j, ż,
GE, GC, CE;
+ GC . * = GE. y + GE.ŷ,
— BG.* = GE.y + GE .j. But since the two curves AE and в have the same curvature at the point E, their abscisses and ordinates have the same fluxions at that point, that is, Ed or is the fluxion both of AD and BG, and de orj is the fluxion both of DE and GE. In the equation above therefore substitute for BG, and y for GE, and it becomes
the radius of curvature, for all curves whatever, in terms of the fluxions of the absciss and ordinate.
75. Further, as in any case either x or y may be supposed to flow equably, that is, either & or ý constant quantities, or or equal to nothing, it follows that, by this supposition, either of the terms in the denominator, of the value of r, may be made to vanish. Thus, when is supposed constant, being then = 0, the value of r is barely
EXAM. 1. To find the radius of curvature to any point
of a parabola, whose equation is ax = y2, its vertex being A, and axis AD.
Now, the equation to the curve being ar=y, the fluxion of it is ax = 2yj; and the fluxion of this again is⋅a* = 2j2, supposing constant; hence then r or
for the general value of the radius of curvature at any point E, the ordinate to which cuts off the absciss AD = x.
Hence, when the absciss r is nothing, the last expression becomes barely žar, for the radius of curvature at the vertex of the parabola; that is, the diameter of the circle of curvature at the vertex of a parabola, is equal to a, the parameter of the axis.
EXAM. 2. To find the radius of curvature of an ellipse, whose equation is a2yc. ax — x2.
Ans. r =
(a2c2 + 4 (a2 — c2) × (ax — x2)
EXAM. 3. To find the radius of curvature of an hyperbola, whose equation is a1y2 = c2. ax + x2.
EXAM. 4. To find the radius of curvature of the cycloid.
Ans. r = 2√ aa
ax, where x is the absciss, and a the diameter of the generating circle.
OF INVOLUTE AND EVOLUTE CURVES.
76. AN Evolute is any curve supposed to be evolved or opened, which having a thread wrapped close about it, fastened at one end, and beginning to evolve or unwind the thread from the other end, keeping always tight stretched the part which is evolved or wound off: then this end of the thread will describe another curve, called the Involute. Or, the same involute is described in the contrary way, by wrapping the thread about the curve of the evolute, keeping it at the same time always stretched..