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(See page 1126, December, 1930, Proceedings)
Lieutenant William S. Parsons, U. S. Navy.—Formation bombing is being given much attention today in gunnery circles and its importance warrants careful study. Let us examine again the figure on page 1127 of the December Proceedings and see how the three families of curves agree, not only with each other, but with what one would expect to be a reasonable prediction.
The left-hand edge of the figure is an excellent place to begin because at this edge the pattern, for a finite target, is zero. Here, then, the three families of curves should meet because the pattern, whether of uniform or tapered density, is zero, and with a zero pattern it is either all or none in the matter of hits.
The fact that the families of curves from 40 per cent to 80 per cent, inclusive, fail to coincide at the left-hand edge makes it necessary to check all the curves in order to determine which are right and which are wrong.
It turns out that the dotted curves are correct within 1 or 2 per cent and that the others in some cases, are as much as 10 per cent in error. A computation made for several points (vertically) in the diagram for the full and the dashed curves at the middle and right-hand edge of the figure shows that the errors at the left-hand edge continue into the figure but are less in amount than at the edge itself. The full curves are correct where they intercept the horizontal axis. Now as to the dotted curves, what do they mean? The author says:
And the curves drawn in dots show the probabilities of at least some parts of the patterns (of whichever type) falling on the target.
In strict theory a uniform pattern of a finite number of bombs must overlap the target a finite amount to make a hit; and a tapered density pattern must overlap it still more. But investigation shows the omission of these effects exercises only a negligible influence on what follows.
From these quotations the dotted curves are defined to mean that the pattern, not necessarily one of the bombs in it, hits the target in the indicated percentages of approaches.
This would all be fine if the number of bombs were infinite. Substitute, for instance, sand for bombs and let our planes drop the sand at the target. Then these curves would give us a fairly accurate picture of the percentage of times that sand would come aboard, because the number of “bombs” would be close enough to infinity for all practical purposes.
However, the practical number of bombs that can be dropped is considerably removed from infinity and further calculations show that with (say) six bombs in the pattern, it does matter considerably whether the pattern is of uniform or of tapered density. In this case, the results for P = 2T, S = T: for uniform density—“some hits” (at least one hit), 90.4 per cent; for tapered density —“some hits” (at least one hit), 82.4 per cent. The curves in the figure referred to give about 77 per cent for both cases.
These differences in results past doubt on the comparison shown on page 1127 relating to the 30 per cent hit cases, particularly since it is positively stated that the patterns under consideration will make “some hits” in the various percentages of attacks. Here the author departs from his previous definition and his statement does not apply to the curves drawn unless the number of
I
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bombs is infinite, when an overlap will constitute “some hits.”
As a further demonstration of the effect of the number of bombs in the pattern, a set of curves is drawn showing the percentage of times that at least one hit is made in the case of a normal pattern to which a tapered density pattern approaches in similarity. A normal distribution of salvo errors is also assumed.
From these curves it is evident that, although the percentage of hits decreases as the mean dispersion in salvo increases (pattern increases in length), the percentage of times that at least one hit is made reaches a different maximum for each value of the number of bombs considered in multiple bomb patterns.1
It is submitted, therefore, that the results obtained from the set of curves above takes into account a determining factor not considered by the author in his paper.
1 The modern shotgun load is an example of a trial and error solution to a problem similar to this one, but much more complicated. To get the maximum weight of lead (percentage of hits) in the ducks in a series of shots, solid lead slugs (zero patterns) should be used, but to reach the “maximum duck” point it is necessary to spread the patterns slightly.