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COR. II. In equation (16) let x = 0; then writing x for h,

F(x) — F(0) =

xn 1.2.3...n

F" (0x);

(20)

of which result (19) is a particular case.

116.] Let us consider some particular cases of these general theorems which arise from particular values of n.

In equation (16) suppose that F'(xo) does not vanish, then

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which gives a symbolical expression for the right-hand member of equation (7), Art. 18, including its residual expression R.

Hence we have the following result. If for all values of x, F(x) = 0, then also r'(x + 0h) = 0; and therefore by (21) F(xo+h) = F(xo). That is, as xo and h are arbitrary, the value of F(x) is the same, whatever is the value of x; and therefore F(x) is constant. The quantity therefore of which the derived function is zero is constant. Hence also conversely we infer that if a derived-function is zero for all values of its subject, then the function of which it is the derived is a constant; and also that a constant is the only continuous quantity of which the derived-function is zero.

Or suppose again that F'(x) = 0, and that "(x) is finite, then n = 2, and

h2 1.2

F(xo+h) F(x) = F" (xo+0h).

(22)

Suppose again, in equation (20), that r (0) = 0, and that r'(0) is finite, then

F(x) = x F(x);

(23)

and therefore every function of a variable x, which vanishes when x = 0, has a for a factor, unless the first derived-function is equal to

when x = 0.

Thus, for example, let F(x)= sin x, which = 0 if x = 0; then its derived-function = cos x, which = 1 when x = 0; therefore x is a factor of sin x.

Similarly, if F(x) = tan x, which

= (sec x)2, which=1 when x =

1

0;

0 if x = 0; then r'(x) therefore x is a factor of tan x.

But if F(x)=e, which = 0 if x = 0, r′(x) =

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2 1

e

x2 2

when x = 0, and is therefore indeterminate; whence

x=

1

we cannot conclude that x is a factor of ex2.

The statement made in most of the ordinary text books on the Theory of Algebraical Expressions, viz. that x is a factor of F(x), if F(x) = 0 when x = 0; or, which is equivalent, that, if F(x) = 0 when x = a, x—a is a factor of F(x), is only one of too many universal propositions which are unduly assumed; it may be and is true in most cases, but it is an unphilosophical desire of generalization which leads us to conclude that it is true in all cases. All possible functions are not known, and cannot be known; and therefore a conscious ignorance ought to guard us from such an error, and be a much more cogent reason against it than the knowledge of a particular instance which disproves the induction; practically however it is found not to be so; let the student be cautious as to such universal assumptions, which rest for proof only on our ignorance.

CHAPTER V.

THE DETERMINATION OF THE ORDERS OF INFINITESIMALS; AND THE EVALUATION OF QUANTITIES WHICH FOR PARTICULAR VALUES OF THE VARIABLES ASSUME THE FORMS

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117.] IN questions such as those proposed for discussion in the present Chapter, it is important to observe what is the exact meaning of the numerical unit; namely, that it is the ratio of equality, and that it is independent of the particular magnitude of the quantities which are compared. Hence it follows, that it is immaterial whether the quantities are infinitesimal, finite, or infinite, provided that we can assure ourselves that they are equal. Whenever therefore the same factor is involved in the numerator and denominator of a fraction, be it of any magnitude and kind, it may be divided out, and the value of the fraction will not be changed by the division: by this means expressions on which operations are to be performed can often be simplified; an instance will illustrate the process and its effect. Suppose that it is required to determine the value of

x-a + (2x2-2ax)
(x2 - a2) +

0

when x = a; in which case the fraction assumes the form but why? Because both numerator and denominator involve a factor (x-a), which is equal to 0 when x = a; but the factor in the numerator being exactly the same as the factor in the denominator, the ratio of one to the other is unity; by which (x − a) may be replaced, and the fraction becomes

therefore

(x − a) $

--

(x − a) + (2x) $
(x + a) $

which is equal to 1, when x = a.

SECTION 1.-The determination of orders of infinitesimals.

118.] It is necessary first to repeat with greater exactness the main points of the account of infinitesimals which was given in rough outline in Art. 8 and 9, Chapter I.

The determination of the Order of Infinitesimals, such as is there assumed, which is a relative term, requires a standard or a base to compare them with. This standard is called the Base of Infinitesimals; and according to the power of the base, which a given infinitesimal expression involves, is its order called.

Thus suppose i to be the base; then, a, b, c, ...... being finite numbers, ai, bi3, ci", are respectively infinitesimals of the first, third, nth orders; i, i are infinitesimals respectively of the one-half and three-fourths orders. Similarly there may be infinitesimals of negative orders.

Suppose that F(i) is a function of infinitesimals, and an infinitesimal itself, whose order is to be compared with i as the base. If F(i) is an infinitesimal of the nth order, i" is a factor of it, and the only other factor that enters with i" is a finite quantity; and of all the terms which contain powers of i entering as factors into their composition, that containing i" is the lowest higher ones being neglected of necessity by virtue of Theorem VI, Art. 9. Therefore, if r(i) is divided by i", the quotient is finite; if it is divided by i raised to an index lower than n, the quotient is infinitesimal; and if it is divided by i raised to an index higher than n, the quotient is infinite. Hence we are enabled to construct the following definition of the order of an infinitesimal expression:

DEF.-Let F(i) be the infinitesimal expression whose order F(i) ir

is to be determined; then, if is infinitesimal for all values

of r less than n, and infinite for all values of r greater than n, F(i) is an infinitesimal of the nth order.

Hence every finite quantity is an infinitesimal of the order 0; for suppose k to be finite, then is infinitesimal for all values

k

ir

of less than 0, and infinite for all values of r greater than 0. Hence, if we use 0 for the general symbol of an infinitesimal, the form, which all finite quantities assume from this point of view, is 0o. This is also evident from the example given in the last Article, where both numerator and denominator = 0, when

x = a; so that x- -a is an infinitesimal which enters into both numerator and denominator, whereby the fraction is of the k x 0

form kx0, k and k' being constants; and as the infinitesimals,

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being exactly equal, are of the same order, the form of the fracand as 0o is equal to 1 in this case, the true value

k

tion is 00 ;

k

of the fraction is

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119.] The results of the preceding Chapter enable us to construct a rule by which this definition may be applied to any function of infinitesimals; and by which the order may be determined.

If we replace x by i in equation (19), Art. 115, we have

F (i)

=

in
1.2.3... (n-1) n

p" (0i).

(1)

i

Now observe the conditions to which F (i) is subject; viz. that r(0) = 0, F'(0) = 0, ...... F-1(0) = 0,

=

and that F" (i) is the derivative, which first does not vanish when 0; and that F(i) and all its derived-functions up to the nth inclusively are continuous and finite for all values of i between i and 0; and observe also the limits, viz. i and 0; but that, as we have to use the equation only when i = 0, we may make the difference between the limits to be infinitesimal.

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Hence

(2)

which is infinitesimal for all values of r less than n, and infinite for all values of r greater than n; and therefore we have the following theorem for the determination of the order of infinitesimals.

THEOREM.-If r(i) is a function of an infinitesimal i, such that F(0) = 0, and that all its derived-functions up to the nth vanish, when i = 0; and that r" (i) does not vanish; then F(i) is an infinitesimal of the nth order, if i is taken to be the infinitesimal-base.

If, when i = O, F(i) assumes an indeterminate form, its value is to be found by the methods which are explained in the following sections of this chapter.

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