where T is the area of a trapezette whose ordinates at successive
distances h are 0, A12φ′ (x1), (A;+A;)¢'(x2), (A=+A; + . . +
A, , 3)¢' (x, ,, , 1), A¢»'fx, ,,); the accents denoting the first differential
coefficient.
85. Volume and Moments of a Briquette.—The application of the methods of §§ 75–79 to calculation of the volume of a briquette leads to complicated formulae. If the conditions are such that the methods of § 61 cannot be used, or are undesirable as giving too much weight to particular ordinates, it is best to proceed in the manner indicated at the end of § 48; i.e. to find the areas of one set of parallel sections, and treat these as the ordinates of a trapezette whose area will be the volume of the briquette.
86. The formulae of § 82 can be extended to the case of a briquette whose top has close contact with the base all along its boundary; the data being the volumes of the minor briquettes formed by the planes x=x0, x=x1, and y=y0, y=y1, . . . The method of constructing the formulae is explained in § 62. If we write
Sp,q, ≡ xpyq u dx dy,
we first calculate the raw values σ0,1, σ1,0, σ1,1, . . . of S0,1, S1,0, S1,1, on the assumption that the volume of each minor briquette is concentrated along its mid-ordinate (§ 44), and we then obtain the formulae of correction by multiplying the formulae of § 82 in pairs. Thus we find (e.g.)
S1,1≅σ1,1
S2,1≅σ2,1−112h2σ0,1
1§ k2rf1, o
52,2-£012 fl? k20'2, o -°{3h2Uo,2'f'r}; ll'k2¢70, o S3»l&U3»l ihzvm
S3,2≅σ3,2 illzvi 12 " k2¢fa,0 'l'4J'5h2k2¢1i,
where om is the total volume of the briquette. 87. If the data of the briquette are, as in § 86, the volumes of the minor briquettes, but the condition as to close contact is not satisfied, we have
if f3;, ';'x1=yvu dx dy = K + L + R - x:', , yg.1, ,, where KExI, ,><qth moment with regard to plane y=0, LE y2Xpth moment with regard to plane x=0, and R is the volume of a bri uette whose ordinate at 's q' (x, ., y,)1 found
by multiplying by PQ xf"“' J/3" the volume of that portion of the original briquette which lies between the planes x=x0, x=x, , y=y0, y=y, . The ordinates of this new briquette at the points of intersection of x=x0, x=x1, . . . with y=y0, y=y1, . . are obtained from the data by summation and multiplication; and the ordinary methods then apply for calculation of its volume. Either or both of the expressions K and L will have to be calculated by means of the formula of § 84; if this is applied to both ex ressions, we have a formula which may be written in a more general)form !b[°u¢(x, y)dxdy =fb f°udxdy . ¢(b, g)
-1* Q rr fr dx dy 1 ff-W d»
irq 3 fl” u dx dy d—¢(33, y) dy
4 ' if dx d ¢l2¢(x|)
+f! 14 y -#dy dxdy.
The second and third expressions on the right-hand side represent areas of trapezettes, which can be calculated from the data; and the fpurtéi expression represents the volume of a briquette, to be calculated in the same way as R above.
88. Cases of Failure.—When the sequence of differences is not such as to enable any of the foregoing methods to be applied, it is sometimes possible to amplify the data by measurement of intermediate ordinates, and then apply a suitable method to the amplified series.
There is, however, a certain class of cases in which no subdivision of intervals will produce a good result; viz. cases in which the top of the figure is, at one extremity (or one part of its boundary), at right angles to the base. The Euler-Maclaurin £ormula (§ 75) assumes that the bounding values of u', u"', . are not infinite; this condition is not satisfied in the cases here considered. It is also clearly impossiblte to express u as an algebraical function of x and y if some value of du/dx or du/dy is to be infinite.
No completely satisfactory methods have been devised for dealing with these cases. One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied. Suppose, for instance, that we require the area of the trapezette ABL in fig. 11; the curve being at right angles to the base AL at A. If QD is the bounding ordinate of one of the component strips, we can calculate the area of QDBL in the ordinary way. The data for the area ADQ are a series of values of u corresponding to equidifferent values of x; if we denote by y the distance of a point
 |
Fig. 11. |
on the arc AD from QD, we can from the series of values of u construct a series of values of y corresponding to equidifferent values of u, and thus find the area of ADQ, treating QD as the base. The process, however, is troublesome.
89. Examples of Applications.—The following are some examples of cases in which the above methods may be applied to the calculation of areas and integrals.
(i) Construction of Mathematical Tables.—Even where u is an explicit function of x, so that 'udx may be expressed in terms of x, it is often more convenient, or construction of a table of values of such an integral, to use finite-difference formulae. The formula/ of § 76 may (see Differences, Calculus of) be written
fzudx = h.;to1.t-1-hi(-ilwlpou-l-»,1215;t¢$'u - .) =;i¢r (hu-fs 62hu +»f'2Jq, f54hu - . Q),
fudx = h.ou+h (, ;146u - 5§ }U6“u +. . .)
= o (hu -1- Q; 6”hu - 5-!;5U5'hu + . .).
The second of these is usually the more convenient. T hus, to. construct a table of values of fudx by intervals of h in x, we first form a table of values of hu for the intermediate values of x, from this obtain a table of values of (1+21162-gi}§ 064+ . .) hu for these values of x, and then construct the table of ffudx by successive additions. Attention must be given to the possible accumulation of errors due to the small errors in the values of u. Each of the above formulae involves an arbitrary constant; but this disappears when we start the additions from a known value. of ” udx.
The process may be repeated. Thus we-have
fzfzudxdx=(o+g;6-5-Hq;6°+...)2h“u ' 1
= (02 +112 " ei'1152 +s15”4'1ru 54 * nights# 'l' - —) hz" = o'(h'u + -112 6'h2u - 515 54h'u + .).
Here there are two arbitrary constants, which may be adjusted in various ways.
The formulae may be used for extending the accuracy of tables, in cases where, if v represents the quantity tabulated, hd-v/dx or h2d"u/dx” can be conveniently expressed in terms of 11 andx to a greater degree of accuracy than it could be found from the table. The process practically consists in using the table as it stands for improving the first or second differences of o and then building up the table afresh.
(ii) Life Insurance.—The use of quadrature-formulae is important in actuarial work, where the fundamental tables are based on experience, and the formulae applying these tables involve the use of) the tabulated values and their differences. 1 », 90. The following are instances of the application of approximative formulae to the calculation of the'volumes of solids.
(i) Timber Measure.—To find the quantity of timber in a trunk with parallel ends, the areas of a few sections must be calculated as accurately as possible, and a formula applied. As the measurements can only be rough, the trapezoidal rule is the most appropriate in ordinary cases. » .
(ii) Gauging.—To measure the volume of a cask, it may be assumed that the interior is approximately a portion of a spheroidal figure. The formula applied) can then be either Sim son s rule or a rule based on Gauss's theorem for two ordinates (§ p 56). In the latter case the two sections are taken at distances i si-I/V 3 = == -2887H from the middle section, where H is the total internal length; and their arithmetic mean is taken to be the mean section of the cask. Allowance must of course be made for the thickness of the wood.
91. Certain approximate formulae for the length of an arc of 8. circle are obtained by methods similar to those of §§ 71 and 79. Let a be the radius of a circle, and 0 (circular measure) the unknown angle subtended by an arc. Then, if we divide θ into m equal parts, and L1 denotes the sum of the corresponding. chords, so that L1=2ma sin (6/2m), the true length of the arc is L1 +ao 3%-§ , +. . ., where ¢=o/sm. similarly, if L, represents the sum of the chords when m (assumed even) is replaced by ém, we have an expression involving L2 and 2¢. The method of, § 71 then shows that, by taking =§ (4L1-Lg) as the Value of the arc, we get rid of terms in ¢“. I we use of to represent the chord of the whole arc, cz the chord of half the arc, and c, the chord of one quarter of the arc, then corresponding to (i) and (iii) of § 70 or § 79'we have 12(8c2-c1) and £g€256c4-4oc2+c;) as approximations to the length of the arc. The first of these is Huygens's rule.
References.—For applications of the prismoidal formula, see Alfred Lodge, Mensuration for Senior Students (1895). Other works on elementary mensuration are G. T. Chivers, Elementary Mensuration (1904); R. W. K. Edwards, Elementary Plane and Solid Mensuration (1902); William H. Jackson, Elementary Solid Geometry (1907); P. A. Lambert, Computation and Mensuration (1907). A. E. Pieroint's Mensuration Formulae (1902) is a handy collection. Rules for calculation of areas are also given in such works as F. Castle, Manual of Practical Mathematics (1903); F. C. Clarke, Practical Mathematics (1907); C. T. Millis, Technical Arithmetic and Geometry
