Page:Optimum height for the bursting of a 105mm shell.pdf/9

 where $$\theta$$ (expressed in radians) is half the angular width of the spray and $$m_*$$ the smallest mass which arrives at $$(x,y)$$ and which is still effective i.e., $$m_*$$ is the mass which has the maximum effective range equal to $$(h^2+x^2+y^2)^{1/2}$$.

Now let $$A$$ denote the effective area of a target. Then the area on the ground sprayed by fragments can be divided, in general, into two regions: An inner region I in which $$\rho$$ is greater than $$A^{-1}$$ and an outer region II in which $$\rho$$ is less than $$A^{-1}$$. Under these circumstances it is clear that in region I where $$\rho\geq A^{-1}$$ we are super effective in the sense that the entire personnel in this region may be expected to be seriously affected. In the other region the probability of hitting a particular target is proportional to $$\rho$$. Thus the number of casualties is given by

where $$\sigma$$ denotes the number of targets per unit area and

In Table IV the density (apart from the factor $$1/4\theta$$) is tabulated for various values of $$h$$. Finally the values of $$E$$ derived on the basis of the results of Table IV are plotted in Fig. 1. It is seen.