1
a = R
W,
the vertical height in feet causing a double expenditure of power is twenty-four; but as above, the whole expense of a double power is increased by only twenty-five per cent.; we should not add one mile for twenty-four feet rise, but one fourth of a mile only, or one mile for each ninety-six feet; and by correcting our former table in this manner, we have the following table:—
| Speed, in miles, per hour. | Equating number. |
|---|---|
| 15 | 88 |
| 20 | 96 |
| 25 | 110 |
| 30 | 128 |
| 40 | 164 |
| 50 | 212 |
| 60 | 268 |
| 100 | 620 |
So much for equating for the ascents. In descending, we have allowed one mile reduction for each mile of twenty-five feet of descending grade; but as in ascending we correct the first made table, so in descending we must also correct as follows. If we needed no steam power either while descending or afterwards, we should only save wood and water; as a general thing the fire must be kept up while descending, and the only gain is a small part of the expense of fuel; so small, in fine, that with the exception of roads which incline for the whole or a great part of their length, no reduction should be made.
COMPARISON OF SURVEYED LINES.
56. The requisite data for an approximate comparison of lines are, the measured length, total rise, total fall.
| Let the length of line A be | 100 | miles, |
| Let the length of line B be | 90 | miles, |
| Whole rise on A | 2000 | feet, |
| Whole rise on B | 5100 | feet, |
| Whole fall on A | 1200 | feet, |
| Whole fall on B | 4300 | feet. |
Assume the number by which to equate, as ninety-six, and we shall have
| Line A. | |
| Ascending, 100 + 2000 96 = | 120.83 |
| Descending, 100 + 1200 96 = | 112.50 |
| Sum | 233.33 |
| Mean | 116.66 |
| Line B. | |
| Ascending, 90 + 5100 96 = | 143.13 |
| Descending, 90 + 4300 96 = | 134.80 |
| Sum | 276.93 |
| Mean | 138.46 |
| The mean equated length of A is | 116.66 |
| The measured length of A is | 100.00 |
| The difference | 16.66 |
| The mean equated length of B is | 138.46 |
| The measured length of B is | 90.00 |
| The difference | 48.46. |
The cost of construction being assumed as the actual length, and that of working as the equated length, we have the final approximate comparison thus:—