Difference, CALCULUS OF FINITE DIFFERENCES.

Difference, CALCULUS OF FINITE DIFFERENCES. Difference implies two quantities of the same kind, and means in arithmetic that quantity which must be added to the smaller in order to produce the larger, and in algebra that quantity which must be added to either to produce the other. Thus if the quantities be the numbers 5 and 7, their arithmetical difference is 2, while their algebraical difference may be either + 2 or - 2. The difference - 2 arises from the fact that we may in algebra ask the question what must be added to 7 to produce 5? and the answer to this is - 2.

In certain groups of problems, chiefly relating to series, differences considered in a particular manner are of peculiar importance, constituting in fact a branch of higher algebra, which took its origin in Brook Taylor's Methodus Incrementorum (1715), and is now called the Method of Differences or the Calculus of Finite Differences. This method we shall briefly illustrate.

Suppose it were required to discover the law of formation, and thence to continue the series of numbers :

4, 3, 0, 1, 12, 39, 88, 165.

It would be wrong to assume that only one law of formation will produce these eight numbers, just as it would be wrong to assume that only one curve could be drawn through eight given points; but for the full discussion of the difficulty here raised the reader must be referred to the chapter on Interpolation in any text-book on the subject. We shall, however, show how to find one law of formation, and use our figures to illustrate the elementary notation of the subject. The process is to take the difference between each term and the succeeding one, and so get the first series of differences, or, as it is called, the series of first differences; the process is repeated on the first differences, and so on, as follows :

No. of term,	1	2	3	4	5	6	7	8
Given Series,	$u_1$	$u_2$	$u_3$	$u_4$	$u_5$	$u_6$	$u_7$	$u_8$
	4	3	0	1	12	39	88	165
1st Differences,	$\Delta u_1$	$\Delta u_2$	$\Delta u_3$	$\Delta u_4$	$\Delta u_5$	$\Delta u_6$	$\Delta u_7$
	-1	-3	1	11	27	49	77
2d Differences,	$\Delta^2 u_1$	$\Delta^2 u_2$	$\Delta^2 u_3$	$\Delta^2 u_4$	$\Delta^2 u_5$	$\Delta^2 u_6$
	-2	4	10	16	22	28
3d Differences,	$\Delta^3 u_1$	$\Delta^3 u_2$	$\Delta^3 u_3$	$\Delta^3 u_4$	$\Delta^3 u_5$
	6	6	6	6	6

The line of third differences suggests a law of formation, and enables us to continue the series as follows :

		165	276	427	624
	77	111	151	197
28	34	40	46
6	6	6	6

It could further be shown that if $u_x$ be the $x$ th term of the series above, then

u_x = x^3 - 7x^2 + 13x - 3.

The operation indicated by $\Delta$ is defined by the following equation, where $u_x$ means any function of $x$ :

\Delta u_x = u_{x+1} - u_x \tag{1}

or, if we denote the 1 added to $x$ by $\Delta x$ , by the equation

\frac{\Delta u_x}{\Delta x} = \frac{u_{x+1} - u_x}{1} \tag{2}

or with the appearance at least of greater generality by

\frac{\Delta u_x}{\Delta x} = \frac{u_{x+h} - u_x}{h} \tag{3}

As a final example let us suppose $u_x = x^2$ . Then we have, using equation (1) above,

\begin{aligned} \Delta u_x &= (x+1)^2 - x^2 \\ &= 2x + 1 \end{aligned}

This is a case of the direct problem of the calculus, but there is also the inverse problem : Of what function is $2x + 1$ the difference ? The solution to this is denoted by the symbols :

z(2x + 1) = x^2,

or, strictly speaking, for reasons which we need not give, by

z(2x + 1) = x^2 + C;

and $x^2 + C$ is said to be the integrate of $2x + 1$ .

Between the Calculus of Finite Differences and the Differential Calculus (see CALCULUS) (a title which means the calculus of infinitesimal differences) there are many important points of contrast and of similarity, which would be not less clearly appreciated if the names were changed, as Boole all but suggested, the former to Calculus of Differences, the latter to Calculus of Limits.

The methods of the Calculus of Differences are in vogue among actuaries and others in dealing with statistics such as mortality tables; and from this calculus are derived many formulæ of approximation of great practical value, such as the rules for finding the area of surfaces bounded by curved lines.