The most obvious application of relation-numbers is to series. Two series may be regarded as equally long when they have the same relation-number. Two finite series will have the same relation-number when their fields have the same cardinal number of terms, and only then—i.e. a series of (say) 15 terms will have the same relation-number as any other series of fifteen terms, but will not have the same relation-number as a series of 14 or 16 terms, nor, of course, the same relation-number as a relation which is not serial. Thus, in the quite special case of finite series, there is parallelism between cardinal and relation-numbers. The relation-numbers applicable to series may be [Pg 56] called "serial numbers" (what are commonly called "ordinal numbers" are a sub-class of these); thus a finite serial number is determinate when we know the cardinal number of terms in the field of a series having the serial number in question. If is a finite cardinal number, the relation-number of a series which has terms is called the "ordinal" number . (There are also infinite ordinal numbers, but of them we shall speak in a later chapter.) When the cardinal number of terms in the field of a series is infinite, the relation-number of the series is not determined merely by the cardinal number, indeed an infinite number of relation-numbers exist for one infinite cardinal number, as we shall see when we come to consider infinite series. When a series is infinite, what we may call its "length," i.e. its relation-number, may vary without change in the cardinal number; but when a series is finite, this cannot happen.