BC, BD, AD, in P, Q, R, S (fig. 9). Then the section of the pyramid
by this plane is the parallelogram PQRS. By drawing Ac and
Ad parallel to BC and BD, so as to meet the plane through
CD in c and d, and producing QP and RS to meet Ac and Ad in q
and r, we see that the area of PQRS is (x/h−x2/h2)× area of
cCDd; this also is a quadratic function of x. The proposition can
then be established for a prismoid generally by the method of § 27 (iv).
The formula is known as the prismoidal formula.
59. Moments.—Since all points on any ordinate are at an equal distance from the axis of u, it is easily shown that the first moment (with regard to this axis) of a trapezette whose ordinate is u is equal to the area of a trapezette whose ordinate is xu; and this area can be found by the methods of the preceding sections in cases where u is an algebraical function of x. The formulae can then be applied to finding the moments of certain volumes.
In the case of the parabolic trapezette, for instance, xu is of degree in x, and therefore the first moment is 13h(x0u0+4x1u1+x2u2). in the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the cross section through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to S0, S2, and 4S1, placed at the extremities and the middle of a line drawn from one end of the solid to the other. The centroid of a hemisphere of radius R, for instance, is the same as the centroid of particles of masses 0, πR2, and 4.34πR2 , placed at the extremities and the middle of its axis; i.e. the centroid is at distance 38R from the plane face.
60. The method can be extended to finding the second, third, . . . moments of a trapezette with regard to the axis of u. If u is an algebraical function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q. may be expressed by a formula λ0v0+y1v1+ . . . λmvm, the qth moment of the trapezette is λ0x0qu0+λ1x1qu1+ . . . λmxmqum, and the mean value of xq is
(λ0x0qu0+λ1x1qu1+ . . . λmxmqum)/(λ0u0+λ1x1+ . . . λmxmqum)
The calculation of this last expression is simplified by noticing that we are only concerned with the mutual ratios of λ0, λ1, . . . and of u0, u1, ., not with their actual values.
61. Cubature of a Briquette.—To extend these methods to a briquette, where the ordinate u is an algebraical function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x＝0 is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y＝0, through the given ordinates of the briquette (§ 44); and that the area of the section is then represented by the ordinate of a trapezette. This ordinate will be an algebraical function of x, and we can again apply a suitable formula.
Suppose, for instance, that u is of degree not exceeding 3 in x, and of degree not exceeding 3 in y, i.e. that it contains terms in x3y3, x3y2, x2y3, &c.; and suppose that the edges parallel to which x and y are measured are of lengths 2h and 3k, the briquette being divided into six elements by the plane x＝x0+h and the planes y＝y0+k, y＝y0+2k, and that the 12 ordinates forming the edges of these six elements are given. The areas of the sides for which x＝x0 and x＝x0 +2h, and of the section by the plane x＝x0+h, may be found by Simpson’s second formula; call these A0 and A2, and A1. The area of the section by a plane at distance x from the edge x＝x0 is a function of x whose degree is the same as that of u. Hince Simpson’s formula applies, and the volume is 13h(A0+4A1+ A2).
The process is simplified by writing down the general formula first and then substituting the values of u. The formula, in the above case, is
13h{38k(u0,0 + 3u0,2 + u0,3) + 4×{38k(u1,0 + . . .) + 38k(u2, + . .)},
where uθ,φ denotes the ordinate for which x＝x0+θh, y＝y0+φk. The result is the same as if we multiplied 38k(v0 + 3v1+3v2+v3) by 13h(u0 + 4u1 +u2), and then replaced u0v0, u0v1, . . . by u0,0, u0,1 . . . The multiplication is shown in the adjoining diagram; the factors 13 and 38; are kept outside, so that the sum u0,0+3u0,1+ . . . +4u1,0+. . . . can be calculated before it is multiplied by 13h, 38k.
| 13×38 | 1 | 4 | 1 |
| 1 | 1 | 4 | 1 |
| 3 | 3 | 12 | 3 |
| 3 | 3 | 12 | 3 |
| 1 | 1 | 4 | 1 |
62. The above is a particular case of a general principle that the obtaining of an expression such as 13h(u0+4u1, +u2) or 38k(v0 +3v1 +3v2+v3) is an operation performed on u0 or v0, and that this operation is the sum of a number of operations such as that which obtains 13hu0 or 38kv0. The volume of the briquette for which u is a function of x and y is found by the operation of double integration, consisting of two successive operations, one being with regard to x, and the other with regard to y; and these operations may (in the cases with which we are concerned) be performed in either order. Starting from any ordinate uθ,φ, the result of integrating with regard to x through a distance 2h is (in the example considered in § 61) the same as the result of the operation 13h(1 + 4E + E2), where E denotes the operation of changing x into x+h (see Differences, Calculus of). The integration with regard to y may similarly (in the particular example) be replaced by the operation 38k(1+3E′+3E′2+E′3), where E′ denotes the change of y into y + k. The result of performing both operations, in order to obtain the volume, is the result of the operation denoted by the product of these two expressions; and in this product the powers of E and of E′ may be dealt with according to algebraical laws.
The methods of §§ 59 and 60 can similarly be extended to finding the position of the central ordinate of a briquette, or the mean qth distance of elements of the briquette from a principal plane.
63. (C) Mensuration of Graphs Generally.—We have next to consider the extension of the preceding methods to cases in which u is not necessarily an algebraical function of x or of x and y.
The general principle is that the numerical data from which a particular result is to be deduced are in general not exact, but are given only to a certain degree of accuracy. This limits the accuracy of the result; and we can therefore replace the figure by another figure which coincides with it approximately, provided that the further inaccuracy so introduced is comparable with the original inaccuracies of measurement.
The relation between the inaccuracy of the data and the additional inaccuracy due to substitution of another figure is similar to the relation between the inaccuracies in mensuration of a figure which is supposed to be of a given form (§ 20). The volume of a frustum of a cone, for instance, can be expressed in terms of certain magnitudes by a certain formula; but not only will there be some error in the measurement of these magnitudes, but there is not any material figure which is an exact cone. The formula may, however, be used if the deviation from conical form is relatively less than the errors of measurement. The conditions are thus similar to those which arise in interpolation (q.v.). The data are the same in both cases. In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates. If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the integral calculus gives the area of the figure.
64. There are three classes of cases to be considered. In the case of mathematical functions certain conditions of continuity are satisfied, and the extent to which the value given by any particular formula differs from the true value may be estimated within certain limits; the main inaccuracy, in favourable cases, being due to the fact that the numerical data are not absolutely exact. In physical and mechanical applications, where concrete measurements are involved, there is, as pointed out in the preceding section, the additional inaccuracy due to want of exactness in the figure itself. In the case of statistical data there is the further difficulty that there is no real continuity, since we are concerned with a finite number. of individuals.
The proper treatment of the deviations from mathematical accuracy, in the second and third of the above classes of cases, is a special matter. In what follows it will be assumed that the conditions of continuity (which imply the continuity not only of u but also of some of its differential coefficients) are satisfied, subject to the small errors in the values of u actually given; the limits of these errors being known.
65. It is only necessary to consider the trapezette and the briquette, since the cases which occur in practice can be reduced to one or other of these forms. In each case the data are the values of certain equidistant ordinates, as described in §§ 43–45. The terms quadrature-formula and cubature-formula are sometimes restricted to formulae for expressing the area of a trapezette, or the volume of a briquette, in terms of such data. Thus a quadrature-formula is a formula for expressing [Ax.u] or ∫udx in terms of a series of given values of u, while a cubature-formula is a formula for expressing [[Vx,y.u]] or ∬udxdy in terms of the values of u for certain values of x in combination with certain values of y; these values not necessarily lying within the limits of the integrations.
66. There are two principal methods. The first, which is the best known but is of limited application, consists in replacing each successive portion of the figure by another figure whose ordinate is an algebraical function of x or of x and y, and expressing the area or volume of this latter figure (exactly or approximately) in terms of the given ordinates. The second consists in taking a comparatively, simple expression obtained in this way, and introducing corrections which involve the values of ordinates at or near the boundaries of the figure. The various methods will be considered first for the trapezette, the extensions to the briquette being only treated briefly.
67. The Trapezoidal Rule.—The simplest method is to replace the trapezette by a series of trapezia. If the data are u0, u1, . . . um, the figure formed by joining the tops of these ordinates is a trapezoid whose area, is h(12u0 +u1 + u2 + . . . +um−1; + 12um). This is called the trapezoidal or chordal area, and will be denoted by C1. If the data are u12, u32, . . . um−12, we can form a series of trapezia by drawing the tangents at the extremities of these ordinates; the sum of the areas of these trapezia will be h(u12+u32;+ . . . +um−12). This is called the tangential area, and will be denoted by T1. The
