The various methods will be considered first for the trapezette, the extensions to the briquette being only treated briefly.
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.
In the case of the briquette the position of the foot of the ordinate u is expressed by co-ordinates x, y, referred to a pair of axes parallel to a pair of sides of the base of the briquette.
The co-ordinates of the edges of the briquette - are (xo, y o), (xo+H, y o), (x o, yo+K), and (xo-1-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.
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.
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 ]y=yo] u 'x' =xo.
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.
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 = o 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.
Suppose, for instance, that u is of degree not exceeding 3 in x, and of degree not exceeding 3 in y, that it contains terms in x3y3, x 3 y 2, 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=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 application of the methods of §§ 75-79 to calculation of the volume of a briquette leads to complicated formulae.
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 = x i,
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.
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 y "`x P u dx dy = K + L + R - X111010-0,0 f xo yo i'?
Y where K-=4, X qth moment with regard to plane y =o, Lm yn X pth moment with regard to plane x =o, and R is the volume of a briquette whose ordinate at (x,.,y s) is found by multiplying by pq x r P - 1 ys 4-1 the volume of that portion of the original briquette which lies between the planes x =xo, y =yo, y = ys.
The ordinates of this new briquette at the points of intersection of x =x 0, x = xi,.
D The second and third expressions on the right-hand side represent areas of trapezettes, which can be calculated from the data; and the fourth expression represents the volume of a briquette, to be calculated in the same way as R above.
A plane parallel to either pair of sides of the briquette is a " principal plane."
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.
The methods of §§ 59 and 60 can similarly be extended to finding the position of the central ordinate of a briquette, or the mean q th of elements of the briquette from a principal plane.
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 quadratureformula 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.