:,'UBATURE 23. The solid figure which corresponds to the trapezette is called a briquette. It is 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. The briquette may usually be regarded as divided into a number of minor briquettes by two sets of parallel planes, the distances between consecutive planes of each set being equal. If the breadth of the briquette is II one way and K the other, and it is divided into m slabs of breadth h by the one set of planes, and into is slabs of breadth k by the other set, then H =mh, K = rnk. The position of an ordinate a is given by its co-ordinates with reference to two planes parallel to the two pairs of bounding planes; if these co-ordinates are x= and y = yo-1-0k, the length of the ordinate may be denoted by ueo.
A process of cubature can usually be regarded as the combina tion of two processes of quadrature, so that cubature-formulae can easily be derived from quadrature-formulae. Suppose, for instance, that m is a multiple of 2 and n is a multiple of 3, and that u is a polynomial of degree not exceeding 3 in x and 3 in y. Then we can group the minor briquettes in sets of 2 X3, as shown in diagram I ; taking one such group, we can regard the area of each section at right angles to the x-axis as found by Simpson's second formula; and, representing each area by an ordinate of a graph, we can use Simpson's formula to find the area of this graph, which represents the volume of the group of 6 briquettes. The result is an expression in which the coefficient
of each u in diagram I is as shown in diagram 2. Or, as an alter native method of getting this result, we can use operators. What we want is the value of a double-integral. We can express the result of integrating with regard to x in the formih( where E denotes the result of changing from x to x+h (see CALCULUS OF DIFFERENCES) ; and the integration with regard to y has the effect of operating with --k(i+3E'-i-3E"1-E"), where E' has a corresponding meaning. The combination of the two opera tions gives the result shown in diagram 2.
The formulae for the corrections of massing (§22) can be ob tained in the same way. Suppose we are dealing with a briquette whose top has close contact with the base all along its boundary, and that we want the moment due to multiplying each element of the volume by xPyq. Let the raw moment obtained by massing the volume of each minor briquette along its mid-ordinate be denoted by N'„. Then the corrected expressions for the moments are given by the formulae