To apply this to any given case we must find the value of l or of k2/a.
Mallet finds these values for the cube, solid and hollow rectangular parallelopipeds, solid and hollow cylinders, &c. In these formulæ we have a direct connection between the dimensions and form of a body and the velocity with which the ground must move beneath it to cause its overthrow.
Fig. 15.
Not only is the case discussed for horizontal forces, but also for forces acting obliquely. Similar reasonings are applied to the productions of fractures in walls, but as there is uncertainty in our knowledge of the co-efficient of force necessary to produce fracture through joints across beds of masonry, the deductions ought not to be applied as the measures of velocity. Where the fractures occur at the base or in horizontal planes, or in those of the continuous beds of the masonry, or through homogeneous bodies, the uncertainty is not so great, and for cases like these Mallet gives several illustrations. The distance to which bodies had been projected, as, for example, ornaments from the tops of pedestals, coping-stones from the edges of roofs, were also used as means of determining the angle at which the shock had emerged, or, if this be known, for determining the velocity.
Thus by a shock in the direction o c, a ball, a, on the top of a pedestal would describe a trajectory to the point c. Let the angle which o c makes with the horizon be e, the vertical height through which the ball has fallen be b, and the horizontal distance of projection be a; then
b = a tan e + a2 /4h cos2 e,
h being the height due to the velocity of projection. Whence
Tan e = 2h ± √4h(h + b) − a2 /a
v2 = a2 g /2 cos2 e (b − a tan e).