CONKERS
Equation A7 can now be evaluated by completing the square in n-space. Write
Y=XP+ULP;
this represents an affine transformation, with Jacobian | P|. Also,
YY’ =(XP+LP")(P'X' + PAL’)
= XPP'X'+ XL! + LX'+ LP’PoL!
= XAX’+ 2LX' + LA“L’,
and Eq. A7 becomes
[7 exp (— YY’ + LAAL)4¥/| P |
= (exp LA“L)/VTA| [7«+» [0° exp (-ut— ++ — vd) dys ++ dyn
so that
{° exp (— KAX' — 2LX") dX = (n¥/2/\/7A |) exp LAL’.
(A8)
Application of Eq. A8 to Eq. A6 leads to Result 2.
Result 2. If a bomb characterized by Eq. A2 is delivered according to the distribution
Eq. Al the expected level of casualties at a point with coordinates (£, v) with respect to the
axes of Eq. Al is given by
[o/V (1 + Aog)(1 + ATR)exp (—(A/2){[E/(1 + Avg) + [v2/(1 + Avg)}).
It may be noted that when og = Tz this becomes a function of the radius V & + v? alone;
and when og = Tg = 0, which represents pin-point bombing,this result reduces to Eq. A2.
POPULATION DISTRIBUTION
Suppose, for concreteness, that a square 20 by 20 miles that contains most of the population of the city has been divided into 100 squares 2 by 2 miles each, and that the nighttime
population within each of these squares has been determined to the nearest thousand.
One might compute the expected level of casualties as follows. Take the center of any
one of these squares andfindits coordinates (£, v) with respect to the assumed bomb pattern,
Eq. Al; then from Result 2 compute the expected level of casualties in that square; finally,
multiply by the number of people in the square. Complete the same processes for each
square, and sum overall 100 squares, to arrive at the total numberof casualties expected.
The problem of several bombs dropped at different points could be treated in like manner.
Such numerical integrations are, however, lengthy and tedious. It would seem natural
j
|
4
to assume normal distributions here in order to eliminate these integrations, but the implications of such a step should be considered.
In the first place such a step does not testify to any sort of conviction about the true
nature of the distribution of the population. It is merely an analytic device that experience suggests may prove applicable within certain wide limits, although results so obtained
could be as bad as the point of worst fit of the city by the surface.
It is only necessary to
choose that point as the aiming point and set A, oz, and Ts = 0; the bomb will always
land at that point andits effect will be felt. only at that point — in short, that worst point
will be the only one to enter into consideration.
Thus if the fitted surface assigns twice
as many people to that point as are actually there, the formulas will “kill” twice as many
people as they should.
ORO-R-17 (App B)
14
CONTFIDENTTAL