making x=0 in this equation, it becomes N=0, and dividing the equation by x, it reduces to 0=P+Qx+ and so on. Hence we

have

...

M=0, N=0, P=0, Q=0...;

in this manner we obtain as many equations as there are co-efficients to be determined.

This principle may be enunciated in another manner, viz.
When an equation of the form

is satisfied by any value given to x, the terms involving the same powers in the two members are respectively equal; for, by transposing all the terms into the second member, the equation will take the form 0=M+Px+Qx2+..., whence

a'—a=0, b'—b=0, c'—c=0 ...

and consequently,

a'=a, b'=b, c'=ı, d'=d... .,

Every equation in which the terms are arranged with reference to a certain letter, and which is satisfied by any value which can be given to this letter, is called an identical equation, in order to distinguish it from a common equation, that is, an equation which can only be satisfied by giving particular values to this letter.

209. The method of indeterminate co-efficients requires that we should know the form of the development, with reference to the exponents of x. The development is generally supposed to be ar. ranged according to the ascending powers of x, commencing with the power o; sometimes, however, this form is not exact; in this case, the calculus detects the error in the supposition.

For example, develop the expression

1

1

Suppose that 3x-2=A+Bx+Cx2+Dx3 + · · · •,

whence, by clearing the fraction, and arranging the terms,

0=-1+3Ax+3B | x2+3C | 23 +3D | 24+...

-A

whence (Art. 208),

-B

-C

-1=0, 3A=0, 3B—A=0.....

Now the first equation, —1=0, is absurd, and indicates that the

above form is not a suitable one for the expression

1

1

X

if we put this expression under the form and suppose

it will become, after the reductions are made,

.),

that is, the development contains a term affected with a negative exponent.

Recurring Series.

210. The development of algebraic fractions by the method of indeterminate co-efficients, gives rise to certain series, called recurring series.

A recurring series is the development of a rational fraction involv ing x, made according to a fixed law, and containing the ascending powers of x in its different terms.

It has been shown in Art. 207, that the development of the ex

which each term is formed by multiplying that which precedes it

b' by-x. α

This property is not peculiar to the proposed fraction; it belongs to all rational algebraic fractions, and it consists in this, viz.: Every rational fraction involving x, may be developed into a series of terms, each of which is equal to the algebraic sum of the products which arise from multiplying certain terms of a particular expression, by certain of the preceding terms of the series.

The particular expression, from which any term of the series may be found, when the preceding terms are known, is called the scale of the series; and that from which the co-efficient may be formed, the scale of the co-efficients.

Clearing the fraction and transposing, we have,

Aa' + Ba' x+Ca' | x2+Da' | x3+Ea' | 204 +...

Whence we perceive that the two first co-efficients are not obtained by any law; but commencing at the third, each co-efficient is formed by multiplying the two which precede it respectively by

and taking the algebraic sum of the products. Hence,

From this law of the formation of the co-efficients, it follows that the third term of the series, Ca2 is equal to

Hence, each term of the required series, commencing at the third, is obtained by multiplying the two terms which precede, respectively by

and taking the sum of the products: hence, this last expression is the scale of the series.

211. Recurring series are divided into orders, and the order is estimated by the number of terms contained in the scale.

The series obtained in the preceding Art. is of the second order. In general, an expression of the form

gives a recurring series of the nth order, the scale of which is

REMARK. It is here supposed that the degree of x in the numerator is less than it is in the denominator. If it was not, it would first be necessary to perform the division, arranging the quantities with reference to x, which would give an entire quotient, plus a fraction similar to the above.