414 
Miscellanea 
He had to find from these data what would be the probable numbers of villages one would 
expect to be attacked no times, once, twice, etc. He used the ordinary summation method, 
that is he calculated the most probable number of infections for each possible combination of 
epidemics. He states that " although the calculation of the probable number of villages to be 
infected in any combination of epidemics is merely a matter of simple arithmetic, yet the 
arithmetic becomes extremely laborious, when the number of epidemics under review is even 
moderately large. Thus in Amritsar District there were six epidemics and a complete evaluation 
of the various combinations of villages attacked never, once, twice, etc., requires the determination 
of 64 distinct products each composed of six terms. Without a mechanical calculator this is an 
impracticable task, and even with such help it is very tedious." 
Two years ago Major Lamb of the Plague Commission (India) gave me the problem of the 
Amritsar District to work out. In order to reduce the labour of the work I devised the 
following process, which, although it contains nothing of mathematical novelty, still attains 
the required result with great economy of labour. In place of the determination of 64 products 
of six terms each, 29 products of two terms each are evaluated, the remainder of the calculation 
being simple addition and subtraction. A mechanical calculator is unnecessary. The time 
employed in the calculation is little over an hour. 
As a simple case let us consider only two probabilities p and q, then, the total frequency 
being taken as unity, the frequency F is 
for 2 times F 2 =p .q = pq 
„ 1 time F 1 =p(l-q) + q (l-p) = (p + q)-^ 
„ 0 times P 0 = (l-p){l-q) =l-(p+q)+ pq. 
Now let P,. = sum of the products of all the probabilities taken r at a time, P 0 being unity. 
Then for the n probabilities, a, b, c, d, e, f, 
(1) ^0 = ^0-^1+ Pi- Ps+ Pi~ P(,+ P*-... 
P,= P 1 -2P 2 +3P 3 -4P i + 5P 5 - 6P e +... 
P 2 = P 2 -3P 3 +6P 4 -10P 6 +15P 6 -... 
F 3 = P 3 -4P 4 + 10P 5 -20P 6 + ... 
P 4 = Pi- 5P 5 + 15P 6 -..., etc. 
and obviously 2*F r =P 0 =l, 
where the coefficients of the P's in the value F r are the same as those in the expansion in powers 
of x, of (1 +A')~( r + 1 ), namely, 
The general formula for F r is thus 
m F-P (r+UP . fr+Dfr + SO p (r+l)(r + 2)(r + 3 ) 
(2) fi r = F r - (9 +1) r" r + 1 H j— J ^ r + 2 1 g g r f + 3+... 
- +{ ' \.2...(n-r) 
A concise and economical method of calculating the values of the P's may be exhibited in the 
following scheme : — 
