For convenience in computation, however, several simple geometrical figures have been assumed as giving with sufficient accuracy the form of the crater of a common mine. See [Pl. XI], Fig. 1. Among these Vauban assumed a cone, ACD, with its vertex at the centre of the charge; Valière a paraboloid of revolution, AHD, with its focus at the centre of the charge; Müller truncated this paraboloid by a horizontal plane through its focus; while Gumpertz and Lebrun adopted the form in common use at their time, and which has been generally accepted since, viz., a frustum of a cone, AEFD, the smaller base of which passes through the centre of the charge and has a radius, EC, equal to one-half the crater radius, AB (or one-half L. L. R., CB).
The volumes of these figures are as follows:
| Vauban’s cone | 1.05 (L. L. R.)3, |
| Valière’s paraboloid | 1.90 (L. L. R.)3, |
| Müller’s truncated paraboloid | 1.84 (L. L. R.)3, |
| The frustum of a cone | 1.83 (L. L. R.)3 = nearly (11/6)(L. L R.)3. |
The cone of Vauban (lately assumed also by Höfer) was abandoned as unsatisfactory, because it did not conform to the craters produced, and, as treated by Höfer, because the charges computed by its use were found to be too small (an error in the wrong direction). The paraboloid of Valière or Müller would seem to conform more nearly to the actual shape assumed by the crater; but it will be observed that the volume of the latter is sensibly the same as that of the truncated cone, and as the volume of earth thrown out is the quantity to be considered, the truncated cone will be assumed as the measure for it.
4. The principle that “the volumes of the craters are proportional to the charges used” is the general statement of the miner’s rule. Assume C and C´ to represent the charges of two mines whose volumes are V and V´, lines of least resistance l and l´, and crater radii r and r´. Assume also that the craters are frustums of cones, the radii of whose larger bases are twice those of the smaller. Then
| C : C´ :: V : V´ :: (11/6)(lr2) : (11/6)(l´r´2), |
or
| C´ = C(V´/V) = C[(11/6)(l´r´2)/(11/6)(lr2)] = C[(l´r´2)/(lr2)] (1) |
Equation (1) is applicable to mines in which r does not differ materially from l or r´ from l´.
From an experimental mine giving a crater of this general type the relations between C, l, and r may be determined, and assuming any two of the quantities C´, l´, and r´ for a mine with a crater nearly similar in form, the other may be found from eq. (1).