Page:EB1911 - Volume 08.djvu/240

 The second part treats of words peculiar to one group. There is no separate glossary of Wallachian.

Of the introduction to the grammar there is an English translation by C. B. Cayley. The dictionary has been published in a remodelled form for English readers by T. C. Donkin.

 DIEZ, a town of Germany, in the Prussian province of Hesse-Nassau, romantically situated in the deep valley of the Lahn, here crossed by an old bridge, 30 m. E. from Coblenz on the railway to Wetzlar. Pop. 4500. It is overlooked by a former castle of the counts of Nassau-Dillenburg, now a prison. Close by, on an eminence above the river, lies the castle of Oranienstein, formerly a Benedictine nunnery and now a cadet school, with beautiful gardens. There are a Roman Catholic and two Evangelical churches. The new part of the town is well built and contains numerous pretty villa residences. In addition to extensive iron-works there are sawmills and tanneries. In the vicinity are Fachingen, celebrated for its mineral waters, and the majestic castle of Schaumburg belonging to the prince of Waldeck-Pyrmont.

 DIFFERENCES, CALCULUS OF (Theory of Finite Differences), that branch of mathematics which deals with the successive differences of the terms of a series.

1. The most important of the cases to which mathematical methods can be applied are those in which the terms of the series are the values, taken at stated intervals (regular or irregular), of a continuously varying quantity. In these cases the formulae of finite differences enable certain quantities, whose exact value depends on the law of variation (i.e. the law which governs the relative magnitude of these terms) to be calculated, often with great accuracy, from the given terms of the series, without explicit reference to the law of variation itself. The methods used may be extended to cases where the series is a double series (series of double entry), i.e. where the value of each term depends on the values of a pair of other quantities.

2. The first differences of a series are obtained by subtracting from each term the term immediately preceding it. If these are treated as terms of a new series, the first differences of this series are the second differences of the original series; and so on. The successive differences are also called differences of the first, second, ... order. The differences of successive orders are most conveniently arranged in successive columns of a table thus:—

Algebra of Differences and Sums. 3. The formal relations between the terms of the series and the differences may be seen by comparing the arrangements (A) and (B) in fig. 1. In (A) the various terms and differences are the same as in § 2, but placed differently. In (B) we take a new series of terms, , , , commencing with the same term &alpha;, and take the successive sums of pairs of terms, instead of the successive differences, but place them to the left instead of to the right. It will be seen, in the first place, that the successive terms in (A), reading downwards to the right, and the successive terms in (B), reading downwards to the left, consist each of a series of terms whose coefficients follow the binomial law; i.e. the coefficients in b &minus; a, c &minus; 2b + a, d &minus; 3c + 3b &minus; a, ... and in &alpha; + &beta;, &alpha; + 2&beta; + &gamma;, &alpha; + 3&beta; + 3&gamma; + &delta;, ... are respectively the same as in y &minus; x, (y &minus; x)², (y &minus; x)³, ... and in x + y, (x + y)², (x + y)³,.... In the second place, it will be seen that the relations between the various terms in (A) are identical with the relations between the similarly placed terms in (B); e.g. &beta; + &gamma; is the difference of &alpha; + 2&beta; + &gamma; and &alpha; + &beta;, just as c &minus; b is the difference of c and b: and d &minus; c is the sum of c &minus; b and d &minus; 2c + b, just as &beta; + 2&gamma; + &delta; is the sum of &beta; + &gamma; and &gamma; + &delta;. Hence if we take &beta;, &gamma;, &delta;, ... of (B) as being the same as b &minus; a, c &minus; 2b + a, d &minus; 3c + 3b &minus; a, ... of (A), all corresponding terms in the two diagrams will be the same.

Thus we obtain the two principal formulae connecting terms and differences. If we provisionally describe b &minus; a, c &minus; 2b + a, ... as the first, second, ... differences of the particular term a (§ 7), then (i.) the nth difference of a is $l - nk + \ldots + (-1)^{n-2}\frac{n\cdot n - 1}{1\cdot 2}c + (-1)^{n-1} nb + (-1)^{n} a,$ where l, k ... are the (n + 1)th, nth, ... terms of the series a, b, c, ...; the coefficients being those of the terms in the expansion of (y &minus; x)n: and (ii.) the (n + 1)th term of the series, i.e. the nth term after a, is $a + n\beta + \frac{n\cdot n - 1}{1\cdot 2}\gamma + \ldots$ where &beta;, &gamma;, ... are the first, second, ... differences of a; the coefficients being those of the terms in the expansion of (x + y)n.

4. Now suppose we treat the terms a, b, c, ... as being themselves the first differences of another series. Then, if the first term of this series is N, the subsequent terms are N + a, N + a + b, N + a + b + c, ...; i.e. the difference between the (n + 1)th term and the first term is the sum of the first n terms of the original series. The term N, in the diagram (A), will come above and to the left of a; and we see, by (ii.) of § 3, that the sum of the first n terms of the original series is $\left(N + na + \frac{n\cdot n - 1}{1\cdot 2}\beta + \ldots \right) - N = na + \frac{n\cdot n - 1}{1\cdot 2}\beta + \frac{n\cdot n - 1\cdot n - 2}{1 \cdot 2 \cdot 3}\gamma + \ldots $

5. As an example, take the arithmetical series a, a + p, a + 2p, ... The first differences are p, p, p, ... and the differences of any higher order are zero. Hence, by (ii.) of § 3, the (n + 1)th term is a + np, and, by § 4, the sum of the first n terms is na + ½n(n &minus; 1)p = ½n{2a + (n &minus; 1)p}.

6. As another example, take the series 1, 8, 27, ... the terms of which are the cubes of 1, 2, 3, ... The first, second and third differences of the first term are 7, 12 and 6, and it may be shown (§ 14 (i.)) that all differences of a higher order are zero. Hence the sum of the first n terms is $n + 7\frac{n\cdot n - 1}{1\cdot 2} + 12\frac{n\cdot n - 1\cdot n - 2}{1\cdot 2\cdot 3} + 6\frac{n\cdot n - 1\cdot n - 2\cdot n - 3}{1\cdot 2\cdot 3\cdot 4} = \tfrac{1}{4}n^{4} + \tfrac{1}{2}n^3 + \tfrac{1}{4}n^2 = {\tfrac{1}{2}n (n + 1)}^2$

7. In § 3 we have described b &minus; a, c &minus; 2b + a, ... as the first, second, ... differences of a. This ascription of the differences to particular terms of the series is quite arbitrary. If we read the differences in the table of § 2 upwards to the right instead of downwards to the right, we might describe e &minus; d, e &minus; 2d + c, ... as the first, second, ... differences of e. On the other hand, the term of greatest weight in c &minus; 2b + a, i.e. the term which has the numerically greatest coefficient, is b, and therefore c &minus; 2b + a might properly be regarded as the second difference of b, and similarly e &minus; 4d + 6c &minus; 4b + a might be regarded as the fourth difference of c. These three methods of regarding the differences lead to three different systems of notation, which are described in §§ 9, 10 and 11.

Notation of Differences and Sums.

8. It is convenient to denote the terms a, b, c, ... of the series by u0, u1, u2, u3, ... If we merely have the terms of the series, un may be regarded as meaning the (n + 1)th term. Usually, however, the terms are the values of a quantity u, which is a function of another quantity x, and the values of x, to which a, b, c, ... correspond, proceed by a constant difference h. If x0 and u0 are a pair of corresponding values of x and u, and if any other value x0 + mh of x and the corresponding value of u are denoted by xm and um, then the terms of the series will be ... un-2, un−1, un, un+1, un+2 ..., corresponding to values of x denoted by ... xn-2, xn−1, xn, xn+1, xn+2....

9. In the advancing-difference notation un+1 &minus; un is denoted by &Delta;un. The differences &Delta;u0, &Delta;u1, &Delta;u2 ... may then be regarded as values of a function &Delta;u corresponding to values of x proceeding by constant difference h; and therefore &Delta;un+1 &minus; &Delta;un denoted by &Delta;&Delta;un, or, more briefly, &Delta;²un; and so on. Hence the table of differences in § 2, with the corresponding values of x and of u placed opposite each other in the ordinary manner of mathematical tables, becomes

The terms of the series of which ... un−1, un, un+1, ... are the first differences are denoted by &Sigma;u, with proper suffixes, so