MENSURATION OF GRAPHS
38. (A) Preliminary.—In § 23 the area of a right trapezium has been expressed in terms of the base and the two sides; and in § 34 the area of a somewhat similar figure, the top having been replaced by an arc of a parabola, has been expressed in terms of its base and of three lengths which may be regarded as the sides of two separate figures of which it is composed. We have now to consider the extension of formulae of this kind to other figures, and their application to the calculation of moments and volumes.
39. The plane figures with which we are concerned come mainly under the description of graphs of continuous variation. Let E and F be two magnitudes so related that whenever F has any value (within certain limits) E has a definite corresponding value. Let u and x be the numerical expressions of the magnitudes of E and F. On any line OX take a length ON equal to xG, and from N draw NP at right angles to OX and equal to uH; G and H being convenient units of length. Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respect, tively. Let KA and LB be the positions of NP corresponding to the extreme values of x. Then the different positions of NP will (if x may have any value from OK to OL) trace out a figure on base KL, and extending from KA to LB; this is called the graph of E in respect of F. The term is also sometimes applied to the line AB along which the point P moves as N moves from K to L.
To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x. It may be shown that this is the same thing as the mean distance
of elements of the graph from an axis through 0 at right angles to OX. Its calculation therefore involves the calculation of the area and the first moment of the graph.
40. The processes which have to be performed in the mensuration of figures of this kind are in effect processes of integration; the distinction between mensuration and integration lies in the different natures of the data. If, for instance, the graph were a trapezium, the calculation of the area would be equivalent to finding the integral, from x=a to x=b, of an expression of the form px+q. This would involve p and q; but, for our purposes, the data are the sides pa+q and pbdq and the base b—a, and the expression of the integral in terms of these data would require certain eliminations. The province of mensuration is to express the final result of such an elimination in terms of the data, without the necessity of going through the intermediate processes.
41. Trapezettes and Briquettes.—A figure of the kind described in § 39 is called a trapezette. A trapezette may therefore be defined as a plane figure bounded by two straight lines, abase at right angles to them, and a top which may be of any shape but is such that every ordinate from the base cuts it in one point and one point only; or, alternatively, it may be defined as the figure generated by an ordinate which moves in a plane so that its foot is always on a straight base to which the ordinate is at right angles, the length of the ordinate varying in any manner as it moves. The distance between the two straight sides, i.e. between the initial and the final position of the ordinate, is the breadth of the trapezette. Any line drawn from the base, at right angles to it, and terminated by the top of the trapezette, is an ordinate of the figure. The trapezium is a particular case.
Either or both of the bounding ordinates may be zero; the top, in that case, meets the base at that extremity. Any plane figure might be converted into an equivalent trapezette by an extension of the method of § 25 (iv).
42. The corresponding solid figure, in its most general form, is such as would be constructed to represent the relation of a magnitude E to two magnitudes F and G of which it is a function; it would stand on a plane base, and be comprised within a cylindrical boundary whose crosssection might be of any shape. We are not concerned with figures of this general kind, but only with cases in which the base is a rectangle. The figure is such as would be produced by removing a piece of a rectangular prism, and is called a briquette. A briquette may therefore be defined as a solid figure bounded by a pair of parallel planes, another pair of parallel planes at right angles to these, a base at right angles to these four planes (and therefore rectangular), and a top which is a surface of any form, but such that every ordinate from the base cuts it in one point and one point only. It may be regarded as generated either by a trapezette moving in a direction at right angles to itself and changing its top but keeping its breadth unaltered, or by an ordinate moving so that its foot has every possible position within a rectangular base.
43. Notation and Definitions.—The ordinate of the trapezette will be denoted by u, and the abscissa of this ordinate, i.e. the distance of its foot from a certain fixed point or origin 0 on the base (or the base produced), will be denoted by x, so that u is some function of x. The sides of the trapezette are the " bounding ordinates "; their abscissae being xo and xo+H, where H is the breadth of the trapezette.
The " midordinate " is the ordinate from the middle point of the base, i.e. the ordinate whose abscissa is xo+2H.
The " mean ordinate " or average ordinate is an ordinate of length l such that Hl is equal to the area of the trapezette. It therefore appears as a calculated length rather than as a definite line in the figure; except that, if there is only one ordinate of this length, a line drawn through its extremity is so placed that the area of the trapezette lying above it is equal to a corresponding area below it and outside the trapezette. Formulae giving the area of a trapezette should in general also be expressed so as to state the value of the mean ordinate (§§ 12 (v), 15, 19).
The " median ordinate " is the ordinate which divides the area of the trapezette into two equal portions. It arises mainly in statistics, when the ordinate of the trapezette represents the relative frequency of occurrence of the magnitude represented by the abscissa x; the magnitude of the abscissa corresponding to the median ordinate is then the " median value of x."
The " central ordinate " is the ordinate through the centroid of the trapezette (§ 32). The distance of this ordinate from the axis of u (i.e. from a line drawn through 0 parallel to the ordinates) is equal to the mean distance (§ 32) of the trapezette from this axis; moments with regard to the central ordinate,are therefore sometimes described in statistics as " moments about the mean."
The data of a trapezette are usually its breadth and either the bounding ordinates or the midordinates of a series of minor trapezettes or strips into which it is divided by ordinates at equal distances. If there are m of these strips, and if the breadth of each is h, so that H =mh, it is convenient to write x in the form xo+Oh, and to denote it by x9, the corresponding value of u being ue. The data are then either the bounding ordinates ue,, ui, . . . um of the strips,
or their midordinates u.'l, 14, . . . 44. the case of the briquette the position of the foot of the
ordinate u is expressed by coordinates x, y, referred to a pair of axes
parallel to a pair of sides of the base of the briquette. If the lengths of these sides are H and K, the coordinates of the angles of the base—i.e. the coordinates of the edges of the briquette—are
(xo, Yo), (xo+H, Yo), (xo, Yo+K), and (xo+H, yo+K).
The briquette may usually be regarded as divided into a series of minor briquettes by two sets of parallel planes, the planes of each set being at successively equal distances. If the planes of one set divide it into m slabs of thickness h, and those of the other into n slabs of thickness k, so that H = nth, K = nk, then the values of x and of y for any ordinate may be denoted by xo+Oh and yofOk, and the length of the ordinate by uo, 0.
The data are usually the breadths H and K and either (i) the edges of the minor briquettes, viz. uo,o, uo,i, . . . ul,o, u1.1, . . . or (ii) the midordinates of one set of parallel faces, viz. uo,i, uo,,i, . 1.44, ... or no, up, . . . up, ..., or (iii) the " midordinates " 14,4, uI,I, . . . ui,I, . . . of the minor briquettes, i.e. the ordinates from the centres of their bases.
A plane parallel to either pair of sides of the briquette is a " principal plane." The ordinate through the centroid of the figure is the " central ordinate."
45. In some cases the data for a trapezette or a briquette are not only certain ordinates within or on the boundary of the figure, but also others forming the continuation of the series outside the figure. For a trapezette, for instance, they may be . . . u_2, u_1, uo, u1 .. .
um, um+1 u,,,+2 ... where Zug denotes the same function of
x=xo+Oh, whether eh lies between the limits o and H or not. These cases are important as enabling simpler formulae, involving central differences, to be used (§ 76).
46. The area of the trapezette, measured from the lower bounding ordinate up to the ordinate corresponding to any value of x, is some function of x. In the notation of the integral calculus, this area is
equal to f xo udx; but the notation is inconvenient, since it implies
a division into infinitesimal elements, which is not essential to the idea of an area. It is therefore better to use some independent notation, such as Ay . u. It will be found convenient to denote
sp(b) 0(a), where 4 (x) is any function of x, by [¢(x)] x _ a; the area of the trapezette whose bounding ordinates are uo and um may then be denoted by [Ay . u] x _ xo or [As . u] 8 = u , instead of by f xo udx.
In the same way the volume of a briquette between the planes x = xo, y =yo, x =a, y = b may be denoted by
[[Vx.y u]y=yo]x=xxo'
47. The statement that the ordinate u of a trapezette is a function of the abscissa x, or that u=f(x), must be distinguished from u=f(x) as the equation to the top of the trapezette.
In elementary geometry we deal with lines and curves, while in mensuration we deal with areas bounded by these lines or curves. The circle, for instance, is regarded geometrically as a line described in a particular way, while from the point of view of mensuration it is a figure of a particular shape. Similarly, analytical plane geometry deals with the curve described by a point moving in a particular way, while analytical plane mensuration deals with the figure generated by an ordinate moving so that its length varies in a particular manner depending on its position.
In the same way, in the case of a figure in three dimensions, analytical geometry is concerned with the form of the surface, while analytical mensuration is concerned with the figure as a whole.
48. Representation of Volume by Area.—An important plane graph is that which represents the volume of a solid figure.
Suppose that we take a pair of parallel planes, such that the solid extends from one to the other of these planes. The section by any intermediate parallel plane will be called a " crosssection." The solid may then be regarded as generated by the crosssection moving parallel to itself and changing its shape, or its position with regard to a fixed axis to which it is always perpendicular, as it moves.
If the area of the crosssection, in every position, is known in terms of its distance from one of the bounding planes, or from a fixed plane. A parallel to them, the volume of the solid can be expressed in terms of the area of a trapezette. Let S be the area of the crosssection at distance x from the plane A. On a straight line OX in any plane take a point N at distance x from 0, and draw an ordinate NP at right angles to OX and equal to S/l, where l is some fixed length (e.g. the unit of measurement). If this is done for every possible value of x, there will be a series of ordinates tracing out a trapezette with base along OX. The volume comprised between the crosssection whose area is S and a consecutive crosssection at distance 0 from it is ultimately SO, when 0 is indefinitely small; and the area between the corresponding ordinates of the trapezette is (S/l) . B = SO/l. Hence the volume of each element of the solid figure is to be found by multiplying the area of the correspond" ing element of the trapezette by 1, and therefore the total volume is 1 X area of trapezette.
The volume of a briquette can be found in this way if the area of the section by any principal plane can be expressed in terms of the distance of this plane from a fixed plane of the same set. The result of treating this area as if it were the ordinate of a trapezette leads to special formulae, when the data are of the kind mentioned in § 44.
49. (B) Mensuration of Graphs of Algebraical Functions.—The first class of cases to be considered comprises those cases in which u is an algebraical function (i.e. a rational integral algebraical function) of x, or of x and y, of a degree which is known.
5o. The simplest case is that in which u is constant or is a linear function of x, i.e. is of the form px + q. The trapezette is then a right trapezium, and its area, if m=l, is +h(uo + u1) or hul.
51. The next case is that in which u is a quadratic function of x, i.e. is of the form px2 + qx + r. The top is then a parabola whose axis is at right angles to the base; and the area can therefore (§ 34) be expressed in terms of the two bounding ordinates and the midordinate. If we take these to be uo and u2, and u1, so that m = 2, we have
62ui,,,+1), .. 'Simpson's second formula is obtained by taking in = 3 and ignoring differences after µ32ulm•
55. The general formulae of § 54 (p being replaced in (i) by +m) may in the same way be applied to obtain formulae giving the area of the trapezette in terms of the midordinates of the strips, the series being taken up to 32fulm or µ32fulm at least, where u is of degree 2f or 2f + 1 in x. Thus we find from (i) that Simpson's second formula, for the case where the top is a parabola (with axis, as before, at right angles to the base) and there are three strips of breadth h, may be replaced by
area = sh(3ui + 2u1 + 31y.
This might have been deduced directly from Simpson's first formula, by a series of eliminations.
56. Hence, for the case of a parabola, we can express the area in terms of the bounding ordinates of two strips, but, if we use midordinates, we require three strips; so that, in each case, three ordinates are required. The question then arises whether, by removing the limitation as to the position of the ordinates, we can reduce their number.
Suppose that in fig. 6 (§ 34) we draw ordinates QD midway between KA and MC, and RE midway between MC and LB, meeting the top in D and E (fig. 8), and join DE, meeting
KA, LB, and MC in H, J, and W. Then it G may be shown that DE is parallel to AB, and that the area of the figure between chord DE and arc DE is half the sum of the areas DHA and EJB. Hence the area of the right trapezium KHJL is greater than the area of the trapezette KACt3L.
If we were to take QD and RE closer to MC, the former area would be still greater. If, on the other hand, we were to take them very close to KA and LB respectively, the area of the trapezette would be the greater. There is therefore some intermediate position such
that the two areas are equal; i.e. such that the area of the trapezette is represented by KL . +(QD + RE).
To find this position, let us Write QM = MR = 0 . KM. Then
WC = 02 . VC, VW = (I — 02) VC;
curved area ACB = j of parallelogram AFGB = aKL . VC;
parallelogram AHJB = KL . VW = (1 — 02) KL . VC.
Hence the areas of the trapezette and of the trapezium will be equal if
i—92=1,0=I/V3.
This value of 0 is the same for all parabolas which pass through D and B and have their axes at right angles to KL. It follows that, by taking two ordinates in a certain position with regard to the bounding ordinates, the area of any parabolic trapezette whose top passes through their extremities can be expressed in terms of these ordinates and of the breadth of the trapezette.
The same formula will also hold (§ 52) for any cubic trapezette through the points.
57. This is a particular case of a general theorem, due to Gauss, that, if u is an algebraical function of x of degree 2p or 2p+I, the area can be expressed in'terms of p + i ordinates taken in suitable positions.
58. The Prismoidal Formula.—It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every crosssection parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's formula may be applied to find the volume of the solid. If the areas of the two ends in the planes K and L are So and S2, and the area of the midsection (i.e. the section by a plane parallel to these planes and midway between them) is S2, the volume Is $H(So + 4S1 + S2), where H is the total breadth.
This formula applies to such figures as the cone, the sphere, the ellipsoid and the prismoid. In the case of the sphere, for instance, whose radius is R, the area of the section at distance x from the centre is ,r(R2—x2), which is a quadratic function of x; the values of So: Sr, and S2 are respectively o, irR2, and o, and the volume is
area = H(uo + 4u1 + u2) = yh(uo + 4u1 + U2).
This is Simpson's formula.
If instead of uo, u1, and u2, we have four ordinates uo, us, U2, and u2, so that m = 3, it can be shown that
area = eh(uo + 3u1 + 3u2 + us).
This is Simpson's second formula. It may be deduced from the formula given above. Denoting the areas of the three strips by A, B, and C, and introducing the middle ordinate uj, we can express A+B;B+C;A+B+C;andBinterms ofuo,U2,u2;u1,U2, u2; uo, ul, u2; and u1, uk, u2 respectively. Thus we get two expressions for A + B + C, from which we can eliminate uI,.
A trapezette of this kind will be called a parabolic trapezette.
52. Simpson's two formulae also apply if u is of the form px2 + qx2 + rx + s. Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the midordinate of the trapezette (with or without this midordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + i. This will be seen by taking the midordinate as the ordinate for which x = o, and noticing that the odd powers of x introduce positive and negative terms which balance one another when the whole area is taken into account.
53. When u is of degree 4 or 5 in x, we require at least five ordinates. If m = 4, and the data are us, u1, u2, u2, u4, we have
area = 15h(7uo + 32U1 + 12u2 + 321,13 + 7u4).
For functions of higher degrees in x the formulae become more complicated.
54. The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences. The breadth of the trapezette being mh, it may be shown that its area is
mh ulm +~4 m2h2u'lm + 1920 m4lZ4ulm `F 322560 moh6uim + 92897280 m2hau l'm + ...
where ulm, ul ,, u i,,,, . . . denote. the values for x = xlm of the
successive differential coefficients of u• with regard to x; the series continuing until the differential coefficients vanish. There are two classes of cases, according as in is even or odd; it will be convenient to consider them first for those cases in which the data are the bounding ordinates of the strips.
(i) If m is even, ulm will be one_of the given ordinates, and we
can express h2ulm, h4u'm, ... in terms of ulm and its even central
differences (see DIFFERENCES, CALCULUS oF). Writing in = 2p, and
grouping the coefficients of the successive differences, we shall find therefore c . 2R . 4?rR2 = tirR2.
area = 2ph up+p232up ' 3p4 — 5p2 34up + To show that the area of a crosssection of a prismoid is of
6 360 the form ax2 + bx + c, where x is
3p8 — 21p4 + 28p2 6 the distance of the section from A B
15120 3 un + • • • one end, we may proceed as in
If u is of degree 2f or 2f + i in x, we require to go up to &21un height , so hg.I h, the case of a pyramid, of , the are area of the e that m must be not less than 2f. Simpson's (first) formula, for by a plane parallel to the base instance, holds for f = i, and is obtained by taking p = i and and at distance x from the vertex
ignoring differences after 32u,,. is clearly x2/h2 X area of base.
(ii) If m is odd, the given ordinates are uo, . . . u;m_l, ulm+;, In the case of a wedge with ... um. We then have parallel ends the ratio x2/h2 is re
m2 — 3 x 3m4 — 50m2 + 135 4 placed by x/h. For a tetrahedron,
area = mh µ141m + 24 µ3 u;m + 5760 µb ulm + two of whose opposite edges are FIG. 9. AB and CD, we require the area
of the section by a plane parallel to AB and CD. Let the distance between the parallel planes through AB and CD be h, and let a plane at distance x from the plane through AB cut the edges AC,
3m6 _ 147m4 + 18131n2 — 4725µ30m.+ . 96768o Where /sub., . . . denote 3(ulm—; + ulm+;), 5(b2ulm_i +
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)X 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
i in x, and therefore the first moment is h(xouo+4xlui+x2u2)• n the case, therefore, of any solid whose crosssection at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, 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 o, TR2, and 4.',r.R2, placed at the extremities
and the middle of its axis; i.e. the centroid is at distance IR from the plane face.
6o. 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
¢+q. may be expressed by a formula Xov0+yivi+ . . . +Xmvm, the qth moment of the trapezette is 71oxOQUO+XixI4U1+ ... +XmxrQum, and the mean value of x4 is
(Xox04u0 + Xix14u1 + ... + Xmxmgum)f(Xouo + Xiul+ ... + XmUm)•
The calculation of this last expression is simplified by noticing that we are only concerned with the mutual ratios of X0, X1, . . . and of us, u1, . . , not with their actual values.
61. Cubature of a Briquette.—To extend these methods to a bri
yette, 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 xo is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y=o, 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 xaya xsy2, x2ys, &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=xo+h and the planes y=yo+k, y=yo+2k, and that the 12 ordinates forming the edges of these six elements are given. The areas of the sides for which x=xo and x=xo+2h, and of the section by the plane x=xo+h, may be found by Simpson's second formula; call these Ao and A2, and A1. The area of the section by a plane at distance x from the edge x=xo is a function of x whose degree is the same as that of u. Hence Simpson's formula applies, and the volume is }h(Ao+4A1+ As).
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
Ih] Ik (uo,o + 3uo,1 + 3uo,s + uo,$) + 4 X Ik(ui,o + ...) + §k (us,o + ..)
where 0,0 denotes the ordinate for which x=xo+9h, y=yo+l'k. The result is the same as if we multiplied
Ik(vo + 3v1+3v2 + vs) by §h(uo + 4u1
+us), and then replaced uovo, uovl, . by uo,o, u0,1 ... The multiplication is shown in the adjoining diagram; the factors I and f are kept outside, so that the sum uo,o+3uo,1+ . . . +4ul,o+... . can be calculated before it is multiplied by ;h . ek.
62. The above is .a particular case of a general principle that the obtaining of an expression such as Ih(uo+4ul+U2) or Ik(vo+3e1+3v2+vs) is an operation performed on uo or vo, and that this operation is the sum of a number of operations such as that which obtains Ihuo or limo. 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 ue,¢, 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 fh(i + 4E + E2), where E `denotes the operation of changing x into x+h (see DIFFERENCES, CALCULUSoF). The integration with regard toy may similarly (in the particular example) be replaced by the operation Ik(1+3E'+3E'2+E's), 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 6o can similarly be extended to finding the position of the central ordinate of a briquette, or the mean q' 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 differentialpcoefficients) 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 §§ 4345. The terms quadratureformula and cubatureformula 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 quadratureformula is a formula for expressing ]As. u] or , fudx in terms of a series of given values of u, while a cubatureformula is a formula for expressing [[Vs,y. u]] or f fudxdy 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 uo, tin ... Um, the figure formed by joining the tops of these ordinates is a trapezoid whose area. is k(}uo+u1+u2+ ...+Um—1+Zumi. This is called the trapezoidal or chordal area, and will be denoted by C1. If the data are uQ, u , . umi, 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 k(ul+ui+... +um— This is called the tangential area, and will be denoted by T1. The
I
I
4
l,XI
3 3
I
3 3
I
3 3
I
4 I2 I2
4
tangential area may be expressed in terms of chordal areas. If we where P, Q, R, . . . satisfy the k equations =I,
write Cf for the chordal area obtained by taking ordinates at P+Q+R+...
intervals ah, then T1=2C;—C1• If the trapezette, as seen from Pp2+Qq2+Rr2+... =o
above, is everywhere convex or everywhere concave, the true area Pp4 Lf Qq4 + Rr4 + .... = o,
lies between C, 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—I in x, according as the data are the bounding ordinates or the midordinates. If A denotes the true area of the original trapezette, and. B the aggregate area of the substituted figures, we have An B, wheredenotes approximate equality. The value of B is found by the methods of §§ 4955. 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 P ah(uo + 4u1 + u2) + ah(u2 + 4u3 + u4) + .. .
f r ah(uo + 4u1 + 2% + 4u3 + 2U4 + ... + 2um2 + 4um1 + um). 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 A th(uo + 3u1 + 3u2+ 2U3 + 3u4 +  • • + 3um_4 + 2um3 +
3um_2 + 3um_1 + um).
(iii) If midordinates are given, and m is a multiple of 3, the repeated application of the formula of § 55 will give
A4jh(3u44+2u;+314 +314;+ ... +2um_l+3umk).
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+26213+1*6'vo+B I, b4v3)=6h(u3+162u3+1 p64u8+~4 g6,u3)
If we replace 54',56u3 in this expression by 5496143, the method of § 68 gives
A11 iah (uo + 5u1 + 142 + 6u3 + u4 + 5us + u6) ;
the expression on the righthand side being an approximate expression for B, and differing from it only by 216H66u3. 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
'Patio + ul+ 142 + 143 + . . . + um_2+ um1 + %um) — ah(auo+ u2 + u4+   +um2+aura)•
Now, if p is any factor of m, there is a series of equidistant ordinates uo, up, u2p, . um—p, u,;,; and the chordal area as determined by these ordinates is
ph(auo+up+ u2p+ . . . . +ump+aura),
which may be denoted by Cp. With this notation, the area as given by Simpson's rule may be written in the form tCl—aC2 or C1+a(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).
A11a(4C1 — C211"C1 + (Cl — C2)•
(ii) m a multiple of 3 (Simpson's second rule).
A 11 1(9C1 — C3) ° Cl + s(C1 — C3).
(iii) m a multiple of 4.
A 1i 1s(64C1 — 2oC2+C4) 11 C1+e(C1 — Co) — a'6(Cl — C4).
(iv) m a multiple of 6 (Weddle's rule, or its repeated application).
A 111o(15C1—6C2+C3) S1 Cl +1(C1 — C2) —?5(C2 — CO.
(v) m a multiple of 12.
A 1' 316(56C1 — 28C2 +803 — C4)
11_ C1+6(Cl — C2) — 6(C2 — C3) + 1i6(Cs — C4).
There are similar formulae in terms of the tangential areas Ti, T2, T3. Thus (iii) of § 68 may be written A 1? kk(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 I) of m, we take
A 2 PCp+ QC4+RC, + .. •
pp2k2 + Qg2k2 + Rr2k2 + ... = 0. The last k 1 of these equations give
I/P : 1/Q : I/R :... = p2(p2 — g2)(p2 — r2) .. . g2(g2—p2)(g2—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 A1PTp+QTQ+RT,.+...
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 tq various objections in practice, such as the following: (i) The assignment of objections coefficients of different ordinates, and even the selection of ordinates for the purpose of finding C2, Co, &c. (§ 70), is troublesome. (ii) This assignment of different coefficients means that different weights are given to different ordinates; and the relative weights may 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 Cl, 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 Cl 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 Et " B
directions of the tangents.
Let KABL (fig. to) 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 lin
and DF and TG perpendicular to LB. ,
in the first place to see whether the erence can
e express
n terms
f
hdif
d
e
Then AD=EB=ah, and the triangles K AVD and BVE are equal. The area of the trapezette is less (in
fig. lo) than the area of the trapezium KABL by twothirds of the area of the triangle ATB (§ 34). This latter area is
End of Article: MENSURATION OF 

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