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Let us suppose there are n of these prismoids each of which is 100 ft.
long, and that beginning with the first end section we denote the successive
cross-sections by the letters A_{0}, A_{1}, A_{2},
A_{3}, ..., A_{n–1} and A_{n}, the subscript of
each A denoting the number of prismoids up to that section and n being
an even number.

It is evident that with L = 100 and treating each A having an *even*
subscript as the end section of a prismoid, 2 × L = 2 × 100
or 200 ft. long, the intermediate As having an odd subscript would correspond
to the A_{m} of Eq. 1, Pg.
62.

Taking then the prismoid whose end sections by this arrangement are
A_{0} and A_{2} and making A_{1}, the middle section,
we have for the volume in cubic yards of the first prismoid

Taking the next prismoid whose end sections are evidently A_{2}
and A_{4}, and whose middle section is A3, and denoting its volume
in cubic yards of V_{2} we have

Similarly for the third prismoid for whose volume we should have

Since the factor (2 × 100) ÷ (6 × 27) is common to all we should have the total volume of n prismoids.

To employ this formula let us suppose that the railroad section whose volume is desired is say 600 feet long. Having selected a scale, say 1 inch to 8 feet, we proceed to plot the consecutive cross-sections, superimposing them one on another as shown in Fig. 3 of Plate IV, and denoting each cross-section by the letter A with the proper subscript showing its number from the first end section.

Since we have plotted our cross-sections to a linear scale of 1 in. = 8 ft. each square inch of actual area of plotted cross-section will represent 8² or 64 square feet, and introducing this factor in the general Eq. 14 we have the particular formula for this scale

In Eq. 17 it is seen that if the entire quantity inside the parenthesis should reduce to an actual area of 1 Sq. Inch the resulting equation would be

It is evident that if we adjust the Planimeter to that Setting which
will give a Reading of 1580.2 Vernier Units when tracing an actual area
of 1 square inch, and then proceed to trace the various plotted cross-sections
continuously and in the manner indicated by the quantities inside the parenthesis
of Eq. 17, the resulting Reading of the Planimeter at the completion of
the entire tracing when multiplied by the value of the Relative Vernier
Unit, which in this case is 0.1, will be the total volume in cubic yards
of the 6 prismoids or the entire total volume of the section between the
end sections A_{0} and A_{6}.

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