NOTHING, DIFFERENCES OR This name is given to certain numbers which are used in so many different theorems, that it is worth while to tabulate them, and to consider them as fundamental numbers of reference. They were first specifically noted in this point of view by the late Bishop Brinkley. We shall here confine ourselves to a description of their derivation, an expeditious mode of calculating them, a table of some of their values, and one of their use.
If we take a series of terms a, b, c, e, &e., and form the successive differences of a [DIFFERENCES, CALCULUS OF], the symbols Pa, &e., have a meaning which refers to the excess of b above a, &c. If then a should happen to be= 0, the symbols is 0, &e., may stand for finite quantities : for instance in But as the preceding series is a set of values of l x(x + 1)-1, in which the first term is 0 gives .61.2-1), it would be necessary in using several series beginning with 0, to make marks of distinction between ei0 in one series, and that in another. The moet useful case is that in which whole and positive powers of 0, 1, 2, 3, Ito., form the eerie* in question : thus if we take the series of cubes, The symbol e," 0*, whenever ta is greater than n, stands for 0; when es is is a it stands for 1 x 2x 3 x .... x ft. In all other cases the differences of 0 "+t may be found from those of 0" by the following equation : = sa 0"1 It ie frequently useful to have the term A' . 0" (r-1). r
arranged in tables. If we wish to make this separately, we have, denoting the preceding fraction by A('). 0" MO Ono = 0" + r The following table contains both the differences, and the differences divided, as just explained, up to those formed from the series of tenth powers; arranged so that simple differences must be looked for above or on the dotted lines, and divided differences below the dotted lines ; the first by means of the left hand column and highest row ; the second by the right hand column and lowest row. Thus The only one not in the table is Al') 0' which is always unity.
The uses of these differences mainly consist in the rapidity with which transformations can be made by means of them, whether of a simple algebraical or of 'a transcendental kind: such as the following, a being a whole number: and so on. [See also OPERATION; SERIES.) The following works contain many properties of these numbers : Herschel, Examples of the Calculus of Finite Differences.' passim; and ' Lib. Useful KnowL : Differential and Integral Calculus,' pp. 253 261, and 307-311.