Now, these quantities will be positive or negative according as the curve is convex or concave towards the axis of abscissas.

d3y . The curves will be convex or concave to the axes, according as is dx2 positive or negative, since by assuming I sufficiently small the sign of the whole series may be made to depend on the signs of these terms.

Or generally, there cannot be a point of contrary flexure unless the first differential, which does not vanish for a particular value of the abscissa be of an odd order.

At a point of contrary flexure a curve from being convex to the axis of abscissas, becomes concave, or vice versa.

The contiguous ordinates of a convex curve are both greater, and of a concave curve both less than the corresponding ordinates to the tangent, but at a point of contrary flexure the ordinates to two points in the curve being near the point of inflexion on each side of it, must be one greater and the other less than the corresponding ordinates to the tangent.

P

Р Q"

T

R

M' M

M"

Let RZ be a plane curve, P a point of inflexion, Tt a tangent at P

AM = x, MM' = MM=h, MP = ƒ (x), M' P'= f (x + h), M" P" = f(x—h)

But at a point of contrary flexure these differences must have a contrary sign,

which cannot be unless

d'y
dx

= 0 or ∞, and if the same value of a which makes

d'y

dx2

d3y

vanish, makes vanish also, then in order that there may be a point of

d'y

dx3

A multiple point is a point in which two or more branches of the curve intersect or touch each other, and is called a double, a triple, &c. point, according to the number of these branches.

Let (A) f (x, y) = 0, be the equation to curve divested of radicals

Let (B) Mp+ N = 0, the first derived equation, then

(1). If the branches of the curve cut one another at the point, there will be several tangents at that point and .. for the value of x and y which belongs to

dy this point, dx

will have as many values as there are branches. Suppose that there are only two branches, and let the two values of p corresponding to these be a, ß. Then by equation (B), we must have

.:. Since

and 6 are supposed to be unequal, we must have M = 0, and .N0, hence by equation (B)

d'y
da

(2). If the branches of the curve touch with a contact of the first order, dy there will be only one value of but there will be several of զ or dx' and in general if the contact be of the nth order, the first n differential co-efficients will have but one value, but the (n + 1)th will have several, we shall in that case have

where M is the same as in equation (B), and L is a rational function of x, y. and the first n differential co-efficients. Hence it may be shown as before, that M = 0, N = 0, and .'.

d" + 1y
dx

The converse, however, does not hold, for it does not follow that these values of a which render

dy
dx

necessarily belong to a multiple point.

Points of the second species where branches of the curve touch are sometimes by way of distinction called osculating points.

To find the first differential co-efficient of the Arc of a Curve considered as a function of the abscissa.

Then it is manifest that the increment PQ of the arc must always be chord PQ and < (PR + RQ), whatever be the values of h.

Now chord PQ = √ h2 + { ƒ (x + h) — ƒ x)}2

Now, series (1), (2), (3) are in the order of their magnitude, whatever be the values of h; .. their first terms are in the order of magnitude, and these are

To find the first differential co-efficient of the Area of a Curve considered as a

function of the abscissa.

Equation to curve

y = f(x)

Area APM A = Q(x)

AM = x, MP = y, MM' h.

Now, it is manifest that MPQ the increment of the area is always parallelogram MPNM' and parallelogram MRQM' whatever be the value

of h.

Now, parallelogram PM' = y. h..........

And series (1), (2), (3), are in the order of their magnitude, whatever be the value of h, and.. their first terms are so; hence

dA
= y
dx

To find the first differential co-efficient of the surface of a solid of Revolution, considered as a function of the abscissa of the generating curve.

Let the surface be generated by the revolution of

curve AZ whose equation, is

y = f (x)

round AX as an axis.

R

Р

Z D

N

Ꮓ Q

Let surface generated by arc AP (= s) be S, and let PQ the increment of arc s be k, S =

(x)

AM = x, MP = y, MM' = h.

Then it is manifest that the increment of the sur

A

M M X

face generated by PQ, is always less than the surface generated by PQ stretched out perpendicular to M'Q from Q, and always greater than the surface generated by PQ stretched out perpendicular to MP from P.

i. e. The surface generated by PQ surface of cylinder rad. = MP, height = k

h

h2

ds h

(2)

Surface of 2nd cylinder = 2x (y+p. '}{+q. 17.2+..) (da · +..)...(3)

dx

And series (1), (2), (3), are in the order of magnitude, whatever be the value of h, and .. their first terms are in order of magnitude; hence

To find the first differential co-efficient of the volume of a solid of Revolution, considered as a function of the abscissa of the generating curve.

Let the solid be generated by the revolution of a curve whose equation is y = f(x) round the axis of x, and its volume = V = 0 (x).

Then every section of the solid made by a plane perpendicular to the axis of x will be a circle.

Let the area of circular plane whose abscissas is x

= A. x + h = A'.

Then the increment of solid is manifestly always by plane A moving parallel to itself through h, and by A' moving parallel to itself through h.

(1). Now, first solid or Ah = πу2 h

than solid generated than the solid generated

And these three series are in the order of their magnitude, whatever be the value of h, and .. their first terms are so; hence we have

dV
dx

METHOD OF LIMITS.

PROPOSITION.-If there be an equation of the form
A+x=B+ y

where A and B are constant quantities, and x and y are susceptible of all degrees of magnitude, then A = B, and x = y.

For if A be not equal to B, let their difference be represented by P

that is, the variables y and x have a constant difference P, and therefore cannot be made less than P, which is contrary to the hypothesis.

This principle is the foundation of the method of limits, which is used extensively in the investigations of the higher geometry, and has been employed by many writers to establish the doctrines of the differential calculus.

DEFINITION.—When a variable quantity by being continually increased or continually diminished, approaches towards a certain fixed quantity, and approaches nearer to this quantity than any assignable difference, but never actually reaches or becomes equal to it, then that fixed quantity is called the LIMIT of the variable quantity.

Thus a circle is the limit of the area of the inscribed and circumscribed polygons. For by continually increasing the number of sides in the polygon, its area will approach nearer to the area of the circle than by any assignable difference, but the sides of the polygon being straight lines, can never actually coincide with the curved perimeter of the polygon, so that the figures should be equal, and, therefore, by the above definition, the circle is the limit of the inscribed and circumscribed polygons.

-

& approach

In like manner if we can make a variable magnitude A another magnitude A which is fixed, so as to render their difference a less than any assignable magnitude, but without their ever becoming actually equal, then the fixed magnitude A is the limit of the variable magnitude A Let us now consider the differential calculus with reference to these princi ples. Let y be a function of x, such that