The Prismoidal Formula. [From Runkle's Mathematical Monthly 1.1 (October 1858): 21-24.] ⁵

BY CHAUNCEY WRIGHT.

The formula (B + 4 B' + B″) l/6, in which B, B', B″ are three equidistant parallel surfaces, sections of a solid, and l the distance between B and B″, is the expression for the solid contents between

B and B″, not only for the prismoid from which it was first derived, but also for several solids of revolution, as the sphere, ellipsoid, &c. What is the extent of its application?

Let X be the axis perpendicular to which the sections are made, and f(x) the area of the section at the distance x from the origin. The problem then is, What function f will fulfil the conditions of the formula?

Let three sections be made through any solid at the distances (x - h), x, (x + h) from the origin. Then h = l/2 and f(x - h), f (x),f (x - h) will be the areas of the sections, and if the formula apply the solid contents between f (x - h) and f (x+ h) will be [f (x - h) + 4 f (x) + f (x + h)]h/3. But the solid contents is also equal to the integral of the differential solid f(x)dx between the limits x - h and x+h.

Equating these two expressions for the solid contents, we have, if the function f fulfils the conditions of the formula,

To find what form of f will satisfy this equation, develop both its members by Taylor’s theorem. The first member becomes,

The second member becomes

Comparing the last members of (A) and (B) we find the equation

which is satisfied only when the terms beyond the second disappear; that is, by functions which have no fourth and higher derivatives. Hence f(x) must be an algebraic expression of positive integral powers not exceeding the third degree.

In general
f(x) = ax3 + bx2 + cx + e = area of section.

The error in applying this formula, when f(x) is of the fourth or fifth degree is

If h is taken so small, that terms containing its fifth power may be neglected, the prismoidal formula may be applied to any solid. Simpson’s rule, which includes a series of prismoidal formulas, is sufficiently accurate when n is so large that (l/n)5 may be neglected.

The prismoid comes within the limits of this formula. For, being a solid bounded by planes, and the sections perpendicular to its sides being dissimilar polygons, but with the same number of sides, it may be decomposed into prismoids with triangular sections. These prismoids are bounded by one plane and two warped surfaces; the plane

surface being one of the surfaces of the original prismoid; the other two being generated by the varying sides of the triangular section. Since these triangles are not similar, their sides must change at dissimilar rates in passing along the axis, or their base and altitude will not change proportionally.

Let b be the base and a the altitude of any triangle at the origin of x; let α and β be their respective rates of change, in passing along x. Then b + βx and a + αx will be the base and altitude at any distance x from the origin, and its area will be 1/2(a + αx)(b+ βx). Hence the area of the whole polygonal section is f(x) = 1/2 Σ [a + αx)(b+ βx) = 1/2 Σ [ab + (aβ + ba)x + α β x2] which is an expression of the second degree, and is therefore one to which the formula applies.

The formula also applies to the solids generated by the revolution of the conic sections about their axes of symmetry, the general equation of these sections being y2 = mx + nx2. The sections of these solids taken perpendicularly to the axis of x are circles, having y for a radius, and πmx + πnx2 = πy2 = f(x) = area, to which also the formula applies.

But if the conic section be revolved about any axis for which the general equation does not hold, then the prismoidal formula does not apply, because the expression for the area of the section f(x) will involve radicals which have fourth and all higher derivatives. The formula applies in like manner to the solid of revolution of the semicubic parabola, of which the equation is y2 = mx3.

The principles involved in this discussion of the prismoidal formula, may be applied to finding a formula, for solid contents, which shall hold good for bodies, of which the sections are expressed in terms of higher degrees than the third; and also for cases in which the sections are taken at unequal intervals.

This discussion is reserved for the next number of the Monthly.