J. McD. Troup and G. D. Maynard 
399 
and for successive terms in a"- if a s be the first term in oy of equation (x), 
a s _(n — 2s + 2)(n - 2s + l) 
o s _i s 2 (m — 2) 3 
n(n - 1) (?i- 2)(/t- 3) . ... 
. • . « x = a 0 x - - v — or 3 = of! y — — - J - (xu). 
(m — 2) 2 2- (w — 2)- 
We proceed now to deal with an example taken from Newsholme's Vital 
Statistics, p. 34-i. In a town with an average of 106,721 houses there were in the 
course of 7 years 3512 cases of Enteric Fever. By means of the above formulae 
we can calculate the mean number of houses that would be affected once, twice, 
thrice, on the supposition that the distribution is a purely chance one, and compare 
them with the observed numbers, viz., 3350, 78 and 2. 
The values of p 1; j5 2) p s to the nearest unit are 
p x = 3398, p 2 =56, p 3 =l. 
If we had a large number of samples of this size and the laws of chance alone 
were at work, the mean number of houses affected twice would be 56. By means 
of the Standard Deviation we can determine what variations from this value may 
reasonably be expected. 
Now er 2 2 = 51 and therefore a., = 7 (to the nearest unit). This gives a probable 
error of 5, and it would therefore be an even chance that in any sample the 
number of houses affected twice would be as many as 61 or as few as 51. 
We now proceed to obtain by a simple method a close approximation to the 
modal values for p 0 , p l} ... p s , &c, and as the values thus obtained are very 
closely the same as those found for the mean, the formula forms a useful method 
for quickly calculating the approximate values of p Q , p x , ... p s , &c. 
To find the values of p 0 , p 1} ... p s , ... which give a maximum value to the 
probability 
m ! n ! 
m n p 0 ip 1 \...p s \...(iiy>i(2\yp>...(s^...' 
The variable part of the above expression is 
p a \ Pl \p 2 l...(l\)Pi (2 l)A (xiii). 
If this is a minimum it will be nearly equal to each of the expressions got by 
substituting successively 
Po - 1, Pi + 2, Pi - 1 for p Q , p u j),; 
p 0 - 2, p 1 + 3, p 3 - 1 for p 0> p 1} p s ; 
Po - 3, + A pt - 1 for p 0 , p u p t ; 
and so on, 
