In studying the theory of formation bombing, one soon faces questions of the best number of planes to use in the bombing group, how far apart it is best to fly them, and whether or not it is best to endeavor to have them spaced equidistantly. But unfortunately all textbooks seem strangely silent on these new problems.
The more complete theory for irregular targets which are variously presented can be built up later. But to simplify the treatment, let us here consider the target as having only one dimension, such as range, for example. Call T the target size, p the pattern size, S the average M.P.I. error, and P the probability of hits; T, p, and Y all being measured along the same axis.
We must study how T, p, S, and P are related. But in order to avoid dealing with quite so many variables, let us eliminate one of them by expressing the pattern size and the average M.P.I. error in terms of the size of the target. Then we need only determine P, the probability of hits, as a function of (p/T) and (S/T).
In a series of two salvos, assume that the pattern and the target are of equal size, and that one of the two patterns lies exactly on the target while the other pattern is two target lengths away. The average M.P.I. error is one target’s size. A single pattern as large as the target would, when located at that distance from the target, make no hits at all. But this series of two salvos obviously will make SO per cent hits.
It is apparent, therefore, in computing hits for an average M.P.I. error (S/T), that we cannot deal simply with that average value so as to compute the average probability in a single simple operation. Instead, we must deal individually with some large number of equally probable values of the M.P.I. error, and then average the resulting series of percentages of hits.
In the following work, forty approximately equally probable values of the M.P.I. error (twenty on each side of the target’s center) were used in every computation. These M.P.I. errors were found by multiplying (S/T) by the ratios which, in the ordinary probability tables, correspond to probabilities of 0.025, 0.075, 0.125... .0.975.
In the figure the curves drawn as full lines show the probabilities of hits for patterns in which bombs are equidistantly spaced, or spaced so as to make the “density uniform throughout the pattern. The curves drawn in dashes show the probabilities of hits for patterns in which the “density” tapers steadily from zero at the ends to a maximum in the center. And the curves drawn in dots show the probabilities of at least some parts of the patterns (of whichever type) falling on the target.1
The curves are instructive in showing that some improvement in hitting is always obtainable by spacing the bombs in the pattern so that the “density” steadily increases from zero at the ends to a maximum in the center. This gain is hardly of importance, however, except when patterns are larger than, and when the average M.P.I. errors are smaller than, the size of the target. So unless we are making well over 30 per cent hits with patterns larger than the target, there seems no real point in attempting to vary the “density” in the different parts of the pattern even if (as seems not to be the case) we could arrange to do it.2
As the curves all slope downward on the right, there is, with the same average M.P.I. error, invariably some improvement in hits obtainable by the use of a smaller pattern. The amount of improvement so effected seems, however, to be considerably less than is generally imagined; and this small improvement in hits is obtainable only at the expense of completely missing the target considerably more frequently than one would with larger patterns.
For example, consider bombing in which about 30 per cent hits are made; or assume an average M.P.I. error just as large as the target. We get 31 per cent, 30 per cent and 29 per cent hits, respectively, with patterns O, 1, and 2 times the size of the target. But these patterns, respectively, will make some hits in 31 per cent, 57 per cent and 77 per cent of the attacks.
“Dependability,” or the probability of making at least some hits on any one attack, is a factor opposed to the making of quite the largest number of hits in the long run. But though it is a factor difficult to weigh numerically against the average percentage of hits which we make, it surely is something that cannot be completely overlooked.
For example, suppose in practical formation bombing that the bomb sight gives an average M.P.I. error as large as the target and that we are able, if we will, to make the pattern only half as large as the target. Would it not then be sounder doctrine to make the pattern twice as large as the target, so sacrificing only about one-thirtieth of the final percentage of hits in order to be able to count on making some hits each time in 77 per cent of the attacks instead of in only 44 per cent of them?
Personally we think there cannot be the slightest doubt that, under such conditions, it would be desirable to use the larger rather than the smaller of these two pattern sizes. Invariably working for the smallest possible pattern size simply “because it will give most hits,” seems disposing of a very complex problem in an extremely incomplete and superficial fashion. The figure will show those who are evolving doctrines for formation bombing exactly how much in the way of greater “dependability” is often obtainable by sacrificing only a certain very small percentage of hits.3 Without dogmatizing further we, therefore, can safely leave it to these curves to tell their own story.
1 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 even slightly more. But investigation shows the omission of these effects exercises only a negligible influence on what follows.
2 Tapered density patterns, however, would not seem at all difficult to have in torpedo fire where “spreads” are used.
3 Making the pattern as much as three times the M.P.I. error seems an outside limit beyond which we should never consider extending the process.