Case 2d. Given the range and inclination of the plane, to find the charge.
Multiply the number found in the above mentioned table opposite the angle or inclination of the plane, in the second column of multipliers, under the head Ascents, or Descents, according as it is above or below the horizon, by the given range; for the range on a horizontal plane at 45°, the charge for which may be found from the tables of ranges.
Upon inclined planes, at any elevation.
There are always two elevations with which any range, (less than the greatest) may be made; and these elevations are always the complements of each other. The greatest range upon a horizontal plane is at 45°; or when the direction bisects the angle formed by the horizontal and vertical plane; also the greatest change upon any plane is made with that direction which bisects the angle between the plane and the zenith; and all other directions which make equal angles with this direction, (on each side of it) will also make equal ranges on the said plane; for the direction that bisects the angle between any plane and the zenith is the same with respect to that plane as the direction at 45° is with respect to the plane of the horizon.
Rules.—1st. The elevation which gives the greatest range on a given ascent is equal to half the sum of 90° added to the ascent.
2d. The elevation which gives equal ranges on a given ascent, are the complements of each other added to the ascent.
3d. The elevation which gives the greatest range on a descent, is equal to half the complement of the descent.
If the range and inclination be given, the least charge that will reach the object, may be found as follows: multiply the tangent of the proper elevation into the proposed range, for the horizontal range whose charge is required.
Table of Amplitudes.
| Degrees. | Ascents. Multip’rs. | Descents. Multip’rs. | ||
|---|---|---|---|---|
| 1st. cl. | 2d. cl. | 1st. cl. | 2d. cl. | |
| 1 | .983 | 1.02 | 1.02 | .983 |
| 2 | .966 | 1.03 | 1.04 | .966 |
| 3 | .949 | 1.06 | 1.05 | .950 |
| 4 | .932 | 1.07 | 1.07 | .932 |
| 5 | .916 | 1.09 | 1.09 | .916 |
| 6 | .900 | 1.11 | 1.11 | .900 |
| 7 | .884 | 1.13 | 1.13 | .884 |
| 8 | .868 | 1.15 | 1.15 | .868 |
| 9 | .852 | 1.18 | 1.17 | .853 |
| 10 | .836 | 1.20 | 1.19 | .836 |
| 11 | .821 | 1.22 | 1.22 | .821 |
| 12 | .805 | 1.24 | 1.24 | .805 |
| 13 | .789 | 1.27 | 1.27 | .789 |
| 14 | .774 | 1.29 | 1.29 | .774 |
| 15 | .758 | 1.32 | 1.31 | .763 |
| 16 | .742 | 1.35 | 1.34 | .745 |
| 17 | .726 | 1.38 | 1.37 | .730 |
| 18 | .711 | 1.40 | 1.39 | .720 |
| 19 | .693 | 1.45 | 1.42 | .704 |
| 20 | .677 | 1.48 | 1.45 | .690 |
| 21 | .660 | 1.52 | 1.48 | .675 |
| 22 | .643 | 1.56 | 1.52 | .662 |
| 23 | .625 | 1.60 | 1.55 | .645 |
| 24 | .607 | 1.64 | 1.58 | .633 |
| 25 | .589 | 1.70 | 1.62 | .617 |
| 26 | .570 | 1.76 | 1.66 | .603 |
| 27 | .550 | 1.82 | 1.69 | .592 |
| 28 | .530 | 1.86 | 1.73 | .578 |
| 29 | .510 | 1.96 | 1.78 | .562 |
| 30 | .488 | 2.05 | 1.82 | .549 |
| 31 | .466 | 2.14 | 1.87 | .534 |
| 32 | .442 | 2.26 | 1.92 | .526 |
| 33 | .418 | 2.41 | 1.97 | .508 |
| 34 | .393 | 2.55 | 2.02 | .495 |
| 35 | .366 | 2.73 | 2.08 | .488 |
| 36 | .338 | 2.96 | 2.13 | .470 |
| 37 | .309 | 3.24 | 2.20 | .455 |
| 38 | .278 | 3.60 | 2.26 | .443 |
| 39 | .245 | 4.09 | 2.33 | .430 |
| 40 | .210 | 4.80 | 2.40 | .417 |
| 41 | .173 | 5.78 | 2.48 | .404 |
| 42 | .134 | 7.46 | 2.56 | .390 |
| 43 | .092 | 10.90 | 2.64 | .380 |
| 44 | .045 | 22.22 | 2.73 | .370 |
| 45 | .000 | infi- nite. | 2.88 | .360 |