Note. The modulus of the hyperbolic logarithms, is 1; and the modulus of the common logarithms, is 43429448190 &c; and, in general, the modulus of any system, is equal to the logarithm of 10 in that system divided by the number 2.3025850929940&c, which is the hyp. log. of 10. Also, the hyp. log. of any number, is in proportion to the com.log. of the same number, as unity or 1 is to 43429&c, or as the number 2.302585&c, is to 1; and therefore, if the common log. of any number be multiplied by 2.302585&c, it will give the hyp. log. of the same number; or if the hyp. log. be divided by 2·302585&c, or multiplied by 43429&c, it will give the common logarithm.

Denoting any proposed number z, whose logarithm is required to be found, by the compound expression

a + x

a

, the fluxion of the number, is, and the fluxion

a

x3%

Then the fluent of these terms give the logarithm of z

a3 at + &c.

Now, for an example in numbers, suppose it were required to compute the common logarithm of the number 2. This will be best done by the series,

a + x

a

=2, gives a = 3.x; conseq.

a

Making =, and =, which is the constant factor for every succeeding term; also,2m = 2 × 43429448190 =·868588964; 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, OR OF CONTRARY FLEXURE IN CURVES.

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

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, x is is a constant quantity.

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to ŷ in a constant ratio, oror j But constant quantities have no fluxion, or their fluxion is equal to nothing; so that in this case, the fluxion of

j

general rule:

is equal to nothing. And hence we have this

72. Put the given equation of the curve into fluxions;

from which find either or

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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

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x

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.

EXAMPLES.

EXAM. 1. To find the point of inflexion in the curve whose equation is ar2 = a'y + x2y.

This equation in fluxions is 2axx = a1ỷ + 2xyx + x2j, x a2 + x2 Then the fluxion of this quantity

which gives=2ax-2xy

made =

O, gives 2xx (ar —xy) = (a2 + x2) × (a* −ży−xỷ);

a2 + x2

=

a2

y

Lastly, this value of being put equal the former, gives

272

y

; and hence 2.x2 = a2

or 3x2 = a2, and x = a√√, the absciss. Hence also, from the original equation,

y

ax2 a2 + x22 flexion sought.

a2

a, the ordinate of the point of in

EXAM. 2. To find the

fined by the equation ay

point of inflexion in a curve deaax2 + xx.

EXAM. 3. To find the point of inflexion in a curve defined by the equation ay a2x+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 conchoid; 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 equally

A

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, Z = AE, and r = CE the radius of curvature; then is Ed=x, dej, and Ee = 2.

Now, by sim. triangles, the three lines Ed, de, Ee,

are respectively as the three therefore

or i, j, ż,

GE, GC, CE;
GC.* = GE.j;

and the flux. of this eq. is GC.+ GC. GE. y + GE.J, or, because GC: BG, it is GC. ።

=

BG.GE.y + GE .j.

But since the two curves AE and BE 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 ory 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 x is supposed constant, being then = 0, the value of r is barely

EXAM. 1. To find the radius of curvature to any point

of