Page:EB1911 - Volume 18.djvu/162

tangential area may be expressed in terms of chordal areas. If we write C_1/2 for the chordal area obtained by taking ordinates at intervals 1/2ℎ, then T₁＝2C_1/2−C₁. If the trapezette, as seen from above, is everywhere convex or everywhere concave, the true area lies between C₁ and T₁.

68. Other Rules for Trapezettes.—The extension of this method consists in dividing the trapezette into minor trapezettes, each consisting of two or more strips, and replacing each of these minor trapezettes by a new figure, whose ordinate 𝑣 is an algebraical function of 𝑥; this function being chosen so that the new figure shall coincide with the original figure so far as the given ordinates are concerned. This means that, if the minor trapezette consists of 𝑘 strips, 𝑣 will be of degree 𝑘 or 𝑘−1 in 𝑥, according as the data are the bounding ordinates or the mid-ordinates. If A denotes the true area of the original trapezette, and B the aggregate area of the substituted figures, we have A≃B, where ≃ denotes approximate equality. The value of B is found by the methods of §§ 49–55. The following are some examples.

(i) Suppose that the bounding ordinates are given, and that 𝑚 is a multiple of 2. Then we can take the strips in pairs, and treat each pair as a parabolic trapezette. Applying Simpson’s formula to each of these, we have

A≃1/3ℎ(𝑢₀ +4𝑢₁ +𝑢₂) + 1/3ℎ(𝑢₂ + 4𝑢₃ +𝑢₄) + . . . ≃ 1/3ℎ(𝑢₀ +4𝑢₁ + 2𝑢₂ + . . . + 2𝑢_𝑚−2 + 4𝑢_𝑚−1 + 𝑢_𝑚)

This is Simpson’s rule.

(ii) Similarly, if 𝑚 is a multiple of 3, the repeated application of Simpson’s second formula gives Simpson’s second rule

A≃3/8ℎ(𝑢₀ + 3𝑢₁ + 3𝑢₂ + 2𝑢₃ + 3𝑢₄ + . . . + 3𝑢_𝑚−4 + 2𝑢_𝑚−3 +3𝑢_𝑚−2 + 3𝑢_𝑚−1 + 𝑢_𝑚).

(iii) If mid-ordinates are given, and 𝑚 is a multiple of 3, the repeated application of the formula of § 55 will give

A ≃3/8ℎ(3𝑢_1/2 + 2𝑢_3/2; + 3𝑢_5/2 + 3𝑢_7/2; + . . . + 2𝑢_𝑚−3/2 + 3𝑢_𝑚−1/2).

69. The formulae become complicated when the number of strips in each of the minor trapezettes is large. The method is then modified by replacing B by an expression which gives the areas of the substituted figures approximately. This introduces a further inaccuracy; but this latter may be negligible in comparison with the main inaccuracies already involved (cf. § 20 (iii)).

Suppose, for instance, that 𝑚＝6, and that we consider the trapezette as a whole; the data being the bounding ordinates. Since there are seven of these, 𝑣 will be of degree 6 in 𝑥; and we shall have (§ 54 (i))

B＝6ℎ(𝑣₃ + 3/2δ²𝑣₃ + 11/20δ⁴𝑣₃ + 41/840δ⁶𝑣₃)＝6ℎ(𝑢₃ + 3/2δ²𝑢₃ + 11/20δ⁴𝑢₃ + 41/840δ⁶𝑢₃).

If we replace 41/840δ⁶𝑢₃ in this expression by 42/840δ⁶𝑢₃, the method of § 68 gives

A ≃3/10ℎ(𝑢₀+ 5𝑢₁ + 𝑢₂ + 6𝑢₃ + 𝑢₄ +5𝑢₅ +𝑢₆);

the expression on the right-hand side being an approximate expression for B, and differing from it only by 1/840δ⁶𝑢₃. This is Weddle’s rule. If 𝑚 is a multiple of 6, we can obtain an expression for A by applying the rule to each group of six strips.

70. Some of the formulae obtained by the above methods can be expressed more simply in terms of chordal or tangential areas taken in various ways. Consider, for example, Simpson’s rule (§ 68 (i)). The expression for A can be written in the form

4/3ℎ(1/2𝑢₀ + 𝑢₁ + 𝑢₂ + 𝑢₃ + . . . 𝑢_𝑚−2 + 𝑢_𝑚−1 +1/2𝑢_𝑚)−2/3ℎ(1/2𝑢₀+𝑢₂ + 𝑢₄ + . . . + 𝑢_𝑚−2 + 1/2𝑢_𝑚)

Now, if 𝑝 is any factor of 𝑚, there is a series of equidistant ordinates 𝑢₀, 𝑢_𝑝, 𝑢_2𝑝, . . . 𝑢_𝑚−𝑝, 𝑢_𝑚; and the chordal area as determined by these ordinates is

𝑝ℎ(1/2𝑢₀ + 𝑢_𝑝 + 𝑢_2𝑝 + . . . . + 𝑢_𝑚−𝑝 1/2𝑢_𝑚),

which may be denoted by C_𝑝. With this notation, the area as given by Simpson’s rule may be written in the form 4/3C₁−1/3C₂ or C₁+1/3(C₁−C₂). The following are some examples of formulae of this kind, in terms of chordal areas.

(i) 𝑚 a multiple of 2 (Simpson’s rule).

A ≃ 1/3(4C₁ − C₂≃C₁ + 1/3(C₁ − C₂)

(ii) 𝑚 a multiple of 3 (Simpson’s second rule).

A ≃ 1/8(9C₁ − C₃) ≃ C₁ +1/8(C₁ − C₃)

(iii) 𝑚 a multiple of 4.

A ≃ 1/45(64C₁ − 2OC₂+C₄) ≃ C₁ + 4/9(C₁ − C₂) − 1/45(C₁ − C₄).

(iv) 𝑚 a multiple of 6 (Weddle’s rule, or its repeated application).

A ≃1/10(15C₁−6C₂+C₃) ≃ C₁ +1/2(C₁ − C₂) −1/10(C₂ − C₃).

(v) 𝑚 a multiple of 12.

A ≃ 1/35(56C₁ − 28C₂ + 8C₃ − C₄)
≃C₁+3/5(C₁ − C₂) − 1/5(C₂ − C₃) + 1/35(C₃ − C₄).

There are similar formulae in terms of the tangential areas T₁, T₂, T₃. Thus (iii) of § 68 may be written A ≃ 1/8(9T₁ − T₃).

71. The general method of constructing the formulae of § 70 for chordal areas is that, if 𝑝, 𝑞, 𝑟, . . . are 𝑘 of the factors (including 1) of 𝑚, we take

A≃PC_𝑝 + QC_𝑞, +RC_𝑟, + . . . ,

where P, Q, R, . . . satisfy the 𝑘 equations

P + Q + R + . . .	＝1,
P𝑝+Q𝑞2+R𝑟2+ . . .	＝0,
P𝑝4 + Q𝑞4 + R𝑟4 + . . . .	＝0,
	: :
P𝑝2𝑘−2 + Q𝑞2𝑘−2 + R𝑟2𝑘−2 + . . .	＝0.

The last 𝑘−1 of these equations give

1/P : 1/Q : 1/R : . . ＝ 𝑝²(𝑝² − 𝑞²)(𝑝²−𝑞²)(𝑞²−𝑟²) . . . : 𝑞²(𝑞²−𝑝²)(𝑞²−𝑟²) . . . : . . . 𝑟²(𝑟² − 𝑝²)(𝑟² − 𝑞²)

Combining this with the first equation, we obtain the values of P, Q, R, . . .

The same method applies for tangential areas, by taking

A ≃PT_𝑝 +QT_𝑞, +RT_𝑟, + . . .

provided that 𝑝, 𝑞, 𝑟, . . . are odd numbers.

72. The justification of the above methods lies in certain properties of the series of successive differences of 𝑢. The fundamental assumption is that each group of strips of the trapezette may be replaced by a figure for which differences of 𝑢, above those of a certain order, vanish (§ 54). The legitimacy of this assumption, and of the further assumption which enables the area of the new figure to be expressed by an approximate formula instead of by an exact formula, must be verified) in every case by reference to the actual differences.

73. Correction by means of Extreme Ordinates.—The preceding methods, though apparently simple, are open to various objections in practice, such as the following: (i) The assignment of different coefficients of different ordinates, and even the selection of ordinates for the purpose of finding C₂, C₃, &c. (§ 70), is troublesome. (ii) This assignment of different coefficients means that different weights are given to different ordinates; and the relative weights ma not agree with the relative accuracies of measurement. (iii) Different formulae have to be adopted for different values of 𝑚; the method is therefore unsuitable for the construction of a table giving successive values of the area up to successive ordinates. (iv) In order to find what formula may be applied, it is necessary to take the successive differences of 𝑢; and it is then just as easy, in most cases, to use a formula which directly involves these differences and therefore shows the degree of accuracy of the approximation.

The alternative method, therefore, consists in taking a simple formula, such as the trapezoidal rule, and correcting it to suit the mutual relations of the differences.

74. To illustrate the method, suppose that we use the chordal area C₁, and that the trapezette is in fact parabolic. The difference between C₁ and the true area is made up of a series of areas bounded by chords and arcs; this difference becoming less as we subdivide the figure into a greater number of strips.

The fact that C₁ does not give the true area is due to the fact that in passing from one extremity of the top of any strip to the other extremity the tangent to the trapezette changes its direction. We have therefore in the first place to see whether the difference can be expressed in terms of the directions of the tangents.

Fig. 10.

Let KABL (fig. 10) be one of the strips, of breadth ℎ. Draw the tangents at A and B, meeting at T; and through T draw a line parallel to KA and LB, meeting the arc AB in C and the chord AB in V. Draw AD and BE perpendicular to this line, and DF and TG perpendicular to LB. Then AD＝EB＝1/2ℎ, and the triangles AVD and BVE are equal.

The area of the trapezette is less (in fig. 10) than the area of the trapezium KABL by two-thirds of the area of the triangle ATB (§ 34). This latter area is

∆BTE − ∆ATD ＝ ∆BTG − ∆ATD＝1/8ℎ² tan GTB − 1/8ℎ² tan DAT.

Hence, if the angle which the tangent at the extremity of the ordinate 𝑢_θ makes with the axis of 𝑥 is denoted by ψ_θ, we have

area from	𝑢0 to 𝑢1＝1/2ℎ(𝑢0 + 𝑢1) − 1/12ℎ2(tan ψ1 − tan ψθ),
area from	𝑢1 to 𝑢2＝1/2ℎ(𝑢1 + 𝑢2) − 1/12ℎ2(tan ψ2 − tan ψ1),
	: :
area from	𝑢𝑚−1 to 𝑢𝑚 ＝1/2ℎ(𝑢𝑚−1 + 𝑢𝑚) − 1/12ℎ2(tan ψ𝑚 − tan ψ𝑚−1);

and thence, by summation,

A＝C₁ − 1/12ℎ²(tan ψ_𝑚 − ψ₀).

This, in the notation of §§ 46 and 54, may be written

A＝C₁ + [− 1/12ℎ²𝑢′] 𝑥＝𝑥_𝑚
𝑥＝𝑥₀.

Since ℎ＝H/𝑚, the inaccuracy in taking C₁ as the area varies as 1/𝑚².

It might be shown in the same way that

A＝T₁+1/24ℎ²(tan ψ_𝑚−ψ₀)＝T₁ + [1/24ℎ²𝑢′]𝑥＝𝑥_𝑚
𝑥＝𝑥₀.

75. The above formulae apply only to a parabolic trapezette Their generalization is given by the Euler-Maclaurin formula