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Quantities of Materials.

I. Volumes from Cross Sections.

2. Volumes of Single Prismoids.

Method of Measurement.

From the above explanation it is evident that to measure the volume in Cu. Yds. of any prismoid with the Planimeter, we plot the two end areas to any desired scale and from them plot the intermediate Section as above described, and as shown in Fig. 1 of Plate IV. Then having adjusted the Planimeter to the Setting given in the Tables for the scale to which we have plotted the Sections, we trace continuously the two end Sections each once and the middle or Mean Section four times, and having taken the total Reading due to the tracing, we multiply it by the Vernier Unit given in the Table for the scale in question. The product of the Reading thus obtained when multiplied by the Relative Vernier Unit will be the desired volume of the Prismoid.

It is evident that in the case of a prismoid having a less length than 100 ft., the operation would be exactly the same except that having obtained the volume of the prismoid in exactly the same manner as though it were 100 ft. long, we simply multiply the volume thus obtained by the actual length of the given prismoid expressed as a per cent. of 100 ft.

Owing to the strict importance of a thoroughly intelligent understanding of the operation just described, it will be best to illustrate it more fully by the following example:

Let us suppose that we desire to measure the volume in Cu. Yds. of a Section of Railroad excavation, the length of the Section being say 73 ft., and that we have the field notes of the Cross-sections at both ends.

Having selected the Scale of 1 inch = 10 ft. as our plotting scale, we proceed to plot both end sections one upon the other, as shown in Fig. 1 of Plate IV, ABCPD being one End Section and ABHNG the other End Section plotted to the scale selected.

We then find by scale the middle points of the lines CH, PN and DG and connect these middle points by the lines FM and ME, as shown.

Looking in the table headed Volumes of Single Prismoids (Table 7), we find that the Setting for the scale of 1 in. = 10 ft. is 307.0. Adjusting the Planimeter to this Setting, and bringing the instrument to a Zero Reading, we place the Pole in the most favorable position relative to the plotted Sections and bring the Tracer to A as the place of beginning for the tracing. Then carefully, and with regard to the rules applying to all cases of Planimeter measurements, we proceed to trace the Section ADPCB. Instead of stopping when we have arrived again at the beginning point, the tracer is carried on past A and the Section AEMFB is carefully traced four times, after which the tracer is made to trace the Section AGNHB once. The entire tracing is performed without stopping and without regard to or the taking of any intermediate Readings, the only watching of the recording wheels being to see if the Recording Wheel has passed the 0 mark and begun a new revolution during the progress of the tracing.

The tracing of the three Sections having been accomplished in the manner described, the Reading of the Planimeter for the entire tracing is then taken. Looking again in the proper table (Table 7), we find the value of the Relative Vernier Unit for the given scale to be .9.

Let us suppose the reading already taken for the entire tracing to be say 12472.0 Units: the volume of the given tracing, were its length 100 ft, would then be 12472 × .9 = 11224.8 Cu. Yds., but since the length of the measured prismoid is only 73 ft., the required volume is evidently only 73 per cent. of what it would be were it 100 ft. long, and hence the volume required is 73 per cent. of 11224.8 or 11224.8 × .73 = 8198.1 Cu. Yds.

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