Ii Moments of Trapezette 20

II. MOMENTS OF TRAPEZETTE 20. The moments of a trapezette with regard to a line L parallel to its ordinates are defined on the same principle as the moments of a lamina. Let L be at a distance X from the axis of u. We suppose the area to be divided into a very large number of very small equal elements dS. To find the rth moment Mr', we multiply each element dS by its (x—X)r and add the results. The limit of this sum when all the elements are made indefinitely small is We need only consider moments about the axis of u, so that X The moments about the central ordinate ("moments about the mean") are denoted by M2, M3, • • • ; the formu lae for obtaining these from M1', M2, M3, . . . are given in §4 2 I . Ordinates Given.—When the data are uo, u2, . . , um, the rth moment is the area of a trapezette whose ordinates are • • • , x;„ u„,. This area is to be found by a quadra ture-formula.

22.

Areas Given.—When, as is usually the case in the theory of statistics, the data are the areas of the strips (of breadth h) of the trapezette, there are two methods of calculating the mo ments of the trapezette. The areas of the in strips are denoted by al, ag, ag, . • • , the total area being • • • (i.) The first method involves an integration by parts. Let Ak be the sum of the areas from al up to inclusive, so that This area is to be found by a quadrature-formula. The method

applies to all cases in which this can be done.

(ii.) The principle of the second method is similar to that of the Euler-Maclaurin theorem. The rth moment of each strip, say the strip from to is split up into two portions, one of which is easily calculated from the data, while the other is of the form —Ikr(x0), 4),.(x) being a certain function of u and of the derivatives of u and of xr. The procedure is as follows: (a) We regard the area of each strip as massed at its mid ordinate, and thus obtain a raw moment We have then to introduce certain corrections.

(b) The simplest case is that of a double-tailed figure, i.e., a figure whose upper boundary has close contact with the base at its extremities. In this case we only require the corrections for massing. The corrected moments are then given by the formulae (NO, of course, being A) (c) If the figure is not double-tailed, we require also the cor rections for abruptness, which consist in adding as described above, to the expressions given in (23).

These corrections are due to W. F. Sheppard, and are known as Sheppard's corrections, though this name is sometimes given to the corrections for massing alone.