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Published online by Cambridge University Press: 18 August 2016
Every mathematical table consists of a series of values of a function corresponding to successive values of the variable, which last series of values forms the argument of the table. Any such table may be constructed therefore, when the function to be tabulated—which may be called the characteristic function—is known, by the evaluation of that function in terms of the successive values of the argument. It is only, however, when the table to be constructed is of limited extent that this method of formation would be employed. If the table be extensive, and especially if the characteristic function be complex, this, which may be called the direct method, would become too laborious, each value when formed in this way also requiring separate verification. In these circumstances the Method of Differences becomes available for the end in view. This method dispenses with all reference to the characteristic function beyond what is necessary for the formation of a few values (which I call fundamental or primitive values), at stated intervals; and in applying it, each value being dependent on the preceding, verification is obtained by the periodical coincidence with those fundamental values of the corresponding terms in the series in course of being formed.
page 62 note * Vol. xi,, pp. 61, 301; and vol, xii,, p. 136.
page 65 note * The process of this problem receives the name of Synthetic Division; and it is easily extended to the case in which the divisor is a polynomial of the form xm+axm–1+bxm–2…See Hutton's Course, by Davies, voL ii., pp. 127 to 130, and 524; also Penny Cyclopædia, Supplement, article “Power.”
page 66 note * The first coefficient remaining constant throughout the operation, it is unnecessary in practice to repeat it on the successive lines.
page 68 note * By a root of a polynomial, as ø(x), is meant a number which, substituted in it for x, causes it to vanish. It is the same thing, therefore, as a root of the equation ø(x)=0.
page 71 note * The demonstration of the above theorem, which is very simple, will be found in any treatise on Professor J. R. work, although the theory of equations. The most accessible of these is probably young's Analysis and Solution of Cubic and Biquadratic Equations. This strictly elementary, goes really a long way into the subject.
page 72 note * It is not true universally, however. In rare cases two or more roots may be situated so closely together that a somewhat perplexing preliminary analysis is requisite to effect their complete separation. But the separation can always be made.
page 74 note * It was repubilshed in the Appendix to the Ladies'Diary for 1838. It mast be admitted to be in an unattractive and needlessly transcendental form.