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Quantities of Materials.

I. Volumes from Cross Sections.

3. Volumes of Continuous Prismoids.

a. Volumes from Plotted Sections.

Example of Measurement.

To make the above description clearer, let us suppose that in the case of the six prismoids of railroad excavation we have been discussing, we have plotted the cross-section to the scale selected, 1 inch = 8 ft., and that Fig. 3 of Plate IV, are the plotted end areas or cross-sections.

Looking in the table headed “VOLS. OF CONTINUOUS PRISMOIDS,” we find that the Setting there given for the scale of 1 inch = 8 ft. is 268.8. Adjusting the Planimeter to this Setting we place the instrument in its most favorable position and bring all the wheels to a zero Reading.

Selecting the point A as a beginning point we then trace the extreme end sections, A0 and A6, each once— the tracing being continuous. At the completion of this tracing the Reading of the wheels is taken which we will suppose in this case to be say 2340 vernier units. Dividing this Reading by 2 gives 2340 ÷ 2 = 1170 which is the value of the expression (A0 + A6) ÷ 2 of Eq. 16.

We then reset the Planimeter to a zero Reading and perform the other operations indicated by the other terms of the quantities within the parenthesis: that is, we carefully and in the manner prescribed for the tracing of all plane areas with the Planimeter trace continuously and in one operation all of the plotted A's which have an even number as a subscript which in this particular case are A0, A2, A4 and A6. Then without stopping or taking any intermediate Reading we proceed to trace twice each plotted A which has an odd number for a subscript which in this case are A1, A3 and A5.

When the tracing as described is completed we take the Reading of the Planimeter which we will say is 24,950 Vernier Units. From the 24,950 Vernier Units which evidently represent the value of the terms A0 + 2 A1 + A2 + 2 A3 + A4 + 2 A5 + A6, we subtract the 1,170 already found which represents the value of the term (A0 + A6) ÷ 2. Hence 24,950 – 1,170 = 23,780 is the total number of Vernier Units due to tracing the cross-sections in the manner indicated.

Looking in Table 8 we find the value of the Relative Vernier Unit to be 2.0 for the scale 1 inch = 8 ft.

Multiplying the total number of Vernier Units by the given value of one Vernier Unit we have 23,780 × 2.0 = 47,560 Cu. Yds., which is the total volume of the six prismoids measured.

In Formula 14 and 15 deduced for the measurement of the volume of any number of continuous prismoids, it will be remembered that the value of L or the length of a single constituent prismoid was taken as 100 ft., and since two such single prismoids were taken as forming a larger prismoid the length of the larger prismoid thus formed was of course 2 L or 200 ft.

In calculating the factors given in Table 8 for the measurement of continuous Prismoids the distance apart of the end Sections has been taken throughout as being 100 ft. as in the example cited.

In rock and similar excavation the cross-sections which form the A's of the Equations Nos. 14 and 15 are frequently taken much closer together— the distance between the cross-sections often being 50 or 25 ft. or even less.

In such cases the measurements are made in exactly the same manner as when they are 100 ft. apart the factors given in the table being used exactly as in the case of the 100 ft. sections and the correction made for the shorter length between cross-sections by simply multiplying the result obtained by using the Table factors by the length between the end sections for the particular case in question expressed as a per cent. of 100 ft.

Thus if the cross-section interval in any case should be say 300 ft. instead of 100 of ft., the final result obtained by using the factors in the Table would be multiplied by 30 ÷ 100 = .3 to obtain the desired volume of prismoids.

It is evident from the above description of this form of measurement that there are no restrictions as to the number of prismoids except when the quantities become too large to keep track of conveniently and with the provision that the number measured by one operation shall be an even number.

Another limiting condition would also evidently be in the more or less confusion and consequent greater liability to error due to plotting too many sections over one another as described. This however can be partially remedied by using different colored inks for the plotting of the various sections.

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