Fig. 18.

may

be considered as infinite. Thus it appears that in each case the true value of the fraction depends only on A and A', the first terms of the series.

The following rule is applicable to every function that can appear under the indeterminate form

Find the first term of each of the ascending series which express the developements of the numerator and denominator when a+h is substituted in them instead of x. Reduce the new function formed of these first terms to its most simple form, and make ho; the results shall be the different values of the proposed function when x is made equal to a.

Example. Suppose the function to be

√x−√ a+ √(x—a)
√(x3-a1)

which, when x=ɑ, becomes - By substituting ath
instead of ≈, and developing the results into series, the
numerator becomes h+
h

h 24/0

minator ✅✔✅24h

+

2/20

term of each series, we have

+ &c. and the deno

Direct

Let a thread be fastened to it at H, and made to pass
along the curve, so as to coincide with it in its whole Method.
extent from H to F. Let the thread be now unlapped
or evolved from the curve, then its extremity F will de-
scribe another curve line FAPP'. The curve HCF is
called the EVOLUTE of the curve FAP; and the curve
FAP is called the INVOLUTE of the curve HCF.

94. From this mode of conceiving the curve to be generated, we may draw the following conclusions.

1st. Suppose PC to be a portion of the thread detached from the evolute, then PC will be a tangent to the evolute at C.

a

2dly. The line PC will be perpendicular to a tangent to the curve FAP at the point P, or will be normal to the curve at that point. For the point P may be considered as describing at the same time an element of the curve FAP, and an element of a circle 9 Pq', whose momentary centre is C, and which has PC for its radius.

3dly. That part of the curve between F and P, which is described with radii all of which are shorter than CP, is more incurvated than a circle described on C as a centre, with a radius equal to CP. And in like manner PP', the part of the curve on the other side of P, which is described with radii greater than PC, is

less incurvated than that circle.

4thly. The circle q P q' has the same curvature as the curve APP' itself has at P: hence it is called an EQUICURVE circle, and its radius PC is called the RADIUS of CURVATURE at the point C.

95. We are now to investigate how the radius of curvature at any point in FAP any proposed curve may be found.

Let AB and BP be the co-ordinates at P any point in the curve, and PC its radius of curvature; and let PC meet AB in E. Put the abscissa AB, the ordinate BP=y, the arch AP=x, the angle AFP (that is, the arch which measures that angle, radius being unity) v, the radius of curvature PC=r. Take P another point in the curve, and let P'C' be the radius of curvature at that point. Let P'C' meet AB in E', and PC in D, and on D as a centre, with a radius=1, describe an arch of a circle, meeting the radii PC, P'C' in m and n. Then the arch PP' will be the increment of %; and since the angle PDP' is the difference of the angles PEA, P'E'A, the arch mn will be the corresponding increment of v.

Suppose now the point P' to approach continually to P, then the points C and D will approach to C, and the ratio of the arch PP', the increment of z, to the

+&c. Taking now the first arch m n the increment of v, will approach to the ratio

of CP to Cm, that is to the ratio of r to I; therefore
the ratio of r to 1 is the limit of the ratio of AP' to m n,
PP'

or r limit of

m n

and passing to the ratio of the

ย

thus we have obtained a formula ex

pressing the radius of curvature, by means of the fluxion
of the arch of the curve, and the fluxion of the angle
which a normal to the curve makes with the line of the

93. Let HC'CF represent a material curve, or mould. abscissas. We proceed to deduce from this formula

and v

(§ 63.), therefore

y

=*/(4x+a), and, putting r

a/

and, putting for the radius of

3 (a + 4x)

-xy

24/a

If in this general expression, we put x=o, we find
a for the radius of curvature at the vertex of
2√α
the curve.

Ex. 2. Suppose the curve to be an ellipse, required
as in the last example.

Putting a and c to denote the two axes, the equation
of the ellipse is a1y=c1(a x-x2). Hence taking the
first and second fluxions, we have 2aayy=c2x (a—2x),
and 2 a3 3 +2 a2y ÿ=-2 c2x2; whence y =
y
*(a—2x), and — ÿ=
_a3y3+c31⁄23
which expressions,
a'y

c2

2 a1y

..

·y=

by substituting the values of y and y, become

*ལྤ
y'

(§ 39.), and sub

the value last found for

ty

therefore,

stituting this value of i in

it becomes

(a-2x)+4(ax-x2)

4(ax—x3) √(ax—xa)

√ (c*a*+ (a°—c®) (4ax—4x2)

We shall now apply these formula to some examples. which expression, when x=0, becomes simply

98. Example 1.-It is required to find the general expression for the radius of curvature of a parabola.

The equation of the parabola is y=a* x3, there curvature at the vertices of the conjugate axis.

PART

Inverse Method.

PART II. THE INVERSE METHOD OF FLUXIONS.

733

Inverse Method.

So, on the contrary, if we have any fluxional equation Inverte of this last form, we may conclude that

106. It is often convenient to denote the fluent of a
fluxional expression without actually exhibiting that
fluent. For this purpose we shall employ the sign,
putting it before the fluxion whose fluent we mean to
denote. Thus, by the expression faa", is to be an-

derstood the fluent of a x" x; and as this fluent has been
ax2+1
found to be
CO-
n+I
in symbols shortly thus,

This expression which we have found for the value
of y in the particular case of yaxi, or
incides with what we might have found by considering
that when y=l. x, it has been shewn (§ 57.), that

+C, we may express this conclusion

Method.