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whence values for p, q, and r would have been obtained, and trial with the seventh and following coefficients would have shown whether they were correct.

518. If the scale of relation consists of 3 terms it involves 2 constants, p and q; and we must have 2 equations to determine p and y. To obtain the first of these we must know at least 3 terms of the series, and to obtain the second we must have one more term given. Thus to obtain a scale of relation involving two constants we must have at least 4 terms given.

If the scale of relation be $1-px-qx^{2}-rx^{3}$, to find the 3 constants we must have 3 equations. To obtain the first of these we must know at least 4 terms of the series, and to obtain the other two we must have two more terms given; hence to find a scale of relation involving 3 constants, at least 6 terms of the series must be given.

Generally, to find a scale of relation involving m constants, we must know at least 2m consecutive terms.

Conversely, if 2m consecutive terms are given, we may assume for the scale of relation

$1-p_{1}x-p_{2}x^{2}-p_{3}x^{3}\ldots -p_{m}x^{m}$.

519. The Sum of n Terms of a Recurring Series. The method of finding the sum is the same whatever be the scale of relation; for simplicity we shall suppose it to contain only two constants.

Let the series be

and let the sum be S; let the scale of relation be $1-px-qx^{2}$; so that for every value of n greater than 1, we have

Now .