To find from these data the relations between charges for overcharged mines, construct Figs. 2 and 2a, ([Pl. XI])
Fig. (2) gives mines with crater radii of 4y and 12y and a common L. L. R. of 4y.
Divide the distance between A and B into four equal parts, and assume the points of division as the extremities of the crater radii of overcharged mines, each of which exceeds the one next smaller by (¼)AB, and all corresponding to a L. L. R. of 4y.
Fig. (2a) gives common mines with lines of least resistance of 4y and 11y. Divide the distance A´B´ also into four equal parts, and assume the points of division as the extremities of the crater radii of common mines each of which exceeds the one next smaller by (¼)A´B´.
Since the charges for the common mines whose lines of least resistance are respectively 4y and 11y are identical with those of the overcharged mines whose crater radii are 4y and 12y respectively, it is assumed that the charges for the intermediate common mines are the same as would be required to produce the corresponding intermediate overcharged mines.
The increment of the crater radius and line of least resistance of any one of these common mines is equal to 7/8 the increment of the crater radius of the corresponding overcharged mine; consequently the charge which gives an overcharged mine whose L. L. R. and crater radius are l´ and r´, respectively, will produce a common mine whose L. L. R. l will be given by the equation
| l = l´ + (7/8)(r´ - l´). (a) |
Since the charge for a common mine is obtained from equation (4), C = C1(11/6)l3, the charge for the overcharged mine will be
| C = C1(11/6)[l´ + (7/8)(r´ - l´]3, |
as above.