398 Chance Distribution of Cases of Disease in Houses 
It is, therefore, the summation in the case when there are (11 — s) balls and (m — 1) 
compartments, and it is to be noticed that p n - s +i , p n -s+2 , & c -, ai 'e all zero, so that 
it is unnecessary to eliminate the factors containing these expressions. 
To find a s 2 , assume 
(p« ~ Psf = Ap s (p s -1) + Bp s +C 
for all values of p s . Then, putting _p s successively = 0, 1, 2, 
p7 = G, 
(i-p s y= b + c, 
(2-J) s Y = 2A + 2B + C, 
.-. A=l, B=l-2p s , C = p s *, 
and therefore 
(Ps - Ps) 2 = p g (p s -l) + (l- 2p s )p s + p s 2 (viii), 
■ •• S[v (p. - PsY] = S [ vPs (p g - 1)] + (1 - 2p s ) S ( vPs ) + p.? 8 ( V ) 
= S [ V p, (p s - 1)] + (l - 2p s )p s S ( v ) + p* S ( v ) 
= S [VPs (Ps - 1)] + (ps -Ps 2 ) ■ S ( V ). 
Now proceeding as in the previous case, we have 
/ m ! 
\p 0 \ Pl \...(p s - 2) !...(! !)* (2 !)**... (a !)?« . 
S[vp s (Ps-l)] = S 
_ m(m - 1) _ / (m - 2) ! \ 
OO 2 \p 0 \p 1 l...(p s -2)l...(l !)^(2!)^...(«I)P«- 2 .../ 
_ m (m - 1) (m - 2)"~ 2S 
~ (si) 2 ' (n-2s)\ W ' 
For the final summation is equivalent to (v), with p s — 2 substituted for p s , m — 2 
for m, and n — 2s for n, the conditions becoming 
Po+Pi + p*+.-.4-,(p g -2)-f-...=m-2, 
Pi + 2/v+ . . . + s (p s - 2) + . . . = n - 2s. 
We have therefore, 
m (m — 1) (m — 2) n ~~ 2s n ! 
(i !) 2 {11 - 2s) I m n 
_ m(m — l) n ~ s nl , ... 
and p s = —pT ^1 — n ( V11 )- 
c s ! (n — 8) ! m n 
It is only necessary to calculate the first term and the others are rapidly 
obtained as follows : 
p s _ m (m - l) n ~ s (s - 1) ! (n - s + 1) ! _ n - s + 1 
p s Z 1 ~~ s ! (n - s) ! m(m- 1 )" _s+1 ~~ * (m - 1) ' 
= '/.,x 3 /„' 5rrt — :d + P* ~P/ (*)» 
