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

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.

Ꮓ

R

Q

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

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

(x)

Then it is manifest that the increment, of the sur

A

M

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 = &

h

h2

ds h

Surface of 2nd cylinder = 2% (y+p. (+q. 11.3+..) (d+..) (3)

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

ds

dx

ds = 2πу dx

= 2xу √1+ p2

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 = Ty2 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

PROPOSITION.-If there be an equation of the form
A+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.

In like manner if we can make a variable magnitude A α approach 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 - O Let us now consider the differential calculus with reference to these princi ples. Let y be a function of ax, such that

Although two quantities be infinitely small, it does not follow that their ratio will be nothing, for

Hence if we represent two infinitely small quantities by dy and dx, their dy

ratio

da may represent any quantity whatever, a result the same as that which we obtained by the consideration of limits.

When a quantity a is infinitely small relatively to a finite magnitude a, the square of x or x2 is infinitely small relatively to x. For the proportion

1:x:x : x2

shows that 2 is involved in x as often as x is involved in unity, that is to say, an infinite number of times. We may demonstrate in the same manner by the proportion

that if a2 is infinitely small relatively to x, the term 23 must be infinitely small relatively to x2. According to this view, infinitesimals are divided into different orders, thus, in the preceding examples, x is an infinitesimal of the first order, a2 is an infinitesimal of the second order, 3 is an infinitesimal of the third order, and so on.

We may remark, that if x is infinitely small relatively to a, it will likewise be infinitely small when multiplied by a finite quantity b. In fact, since ☛ may be considered as a fraction whose denominator is infinity, we may represent a by, but whether we have or these quantities will equally be nothing relatively to a.

bp

Since an infinitesimal of the first order must be disregarded when connected with a finite quantity, which it cannot increase, so, in like manner, an infinitesimal of the second order must be disregarded when connected with an infinitesimal of the first order, and so on.

The product of two infinitesimals x and y, of the first order, is an infinitesimal of the second order; for from the product zy we derive the proportion

1: yx: xy

which shows, that since y is infinitely small relatively to unity, ay will be infinitely small relatively to x; that is to say, ay will be an infinitesimal of the second order.

In like manner, we might prove that the product of three infinitesimals of the first order, is an infinitesimal of the third order.

The differential calculus may be deduced from the theory of infinitesimals. This method of considering the subject is less philosophical than either of the preceding but the results are precisely the same, and as the principles employed will greatly abbreviate many of the processes of the integral calculus, we shall briefly explain their application.

Let y be a function of x, such that

y = ax

(1)

Let x be increased by an infinitely small quantity which we shall represent by da, and let the corresponding infinitely small increment of y be represented

by dy. Hence when becomes x + dx, y will become y + dy, and we shall have from the above equation

y+dya (x + dx)

.. By (1)

= ax + adx dyadx

(2)

The quantity dy is called the differential of y, and the quantity dr is called the differential of x.

If we divide both sides of equation (2) by dx, we shall have

dy
= α
dx

But we know that a is the differential co-efficient of ar; hence it appears, in the present case, that the first differential co-efficient is the same thing as the ratio of the infinitely small increment of y to the infinitely small increment of x, and that the differential of y is equal to its first differential co-efficient multiplied by the differential of x.

Again, let it be required to find the differential of a function of x, such as ax3.

Let

y = ax3

........

(1)

Let x become + dx, and let the corresponding change in y be represented by y + dy.

But a (dx)3 being an infinitesimal of the third order, cannot augment 3a (dx)2, and may therefore be rejected, and in like manner 3a (dx)2 being an infinitesimal of the second order, cannot augment 3ax2 dx, and may therefore be rejected, so that equation (2) is reduced to

but 3ax is the differential co-efficient of ax3, so that in this case also the differential co-efficient is the same as the ratio of the infinitely small increments of y and x, and the differential of y is equal to its first differential .co-efficient multiplied by the differential of x.

Generally, let

y = f (x)

let x become x + dx and y become y + dy

y + dy = f (x + dx)

But (dx), (dx)3,

.. being infinitesimals of the second, third, . . . orders, cannot augment Adx, and may therefore be rejected, hence the above equation becomes

dy Adx

BBB

but according to the principles which we have already explained, A is the first differential co-efficient of y or ƒ (r), hence it appears that

If y be a function of x, the first differential co-efficient of y may be considered as the ratio of the differentials, or infinitely small increments of y and x; and the differential of y is always equal to the first differential co-efficient of y, multiplied by the differential of x.

z,

each

In order to find the differential of the product of two variables u and of which is a function of x, we shall suppose that when x becomes x + dr, becomes udu, and z becomes z + dz.

But du da being an infinitesimal of the second order, may be neglected.

dy udz + zdu

which agrees with the result already found by a different process.

To find the differential of sin. x according to this method.

Let us now show how we may resolve the problem of tangents by the method of infinitesimals.

P'Q which is the infinitely small increment of y will be represented by dy

PQ = MM'

x

dr