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

(5)$N=EA\sigma$

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

(6)$E=A^{-1}\iint _{\mathrm {region} \,I}dxdy+\iint _{\mathrm {region} \,II}\rho (x,y)\,dxdy.$

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 that within the uncertainties of the problem the optimum height is about 75 ft. for a wide range of target areas.