J. McD. Troup and G. D. Maynard 397 
In the above expression it should be noted that both series of factors in the 
denominator can be carried to infinity, for as there are only n balls, p n+1 , p n +2, 
&c, are all zero, and therefore p n+l ! = 1, (n + 1 l) Pf >+' 1 = 1, &c. The importance of 
this will appear later. 
Til ! 
Now let ?7 = '- — (ii). 
Then r\ is the general term in the expansion of 
x- X s \ m 
(l + «. + f] + . •• + -, + •..) 
for all positive integral values of p 0 , p lt p 2 , &c, subject to the conditions : 
p 0 + Pi+p->+ ... +p s + ■ ■ ■ = m (iii), 
p l + 2p, + . . . + rp r + ... =n (iv), 
and 8 (rj) = coeft. of a,' 1 in f 1 + x + ^- + . . . + — + . 
= coeft. of x 11 in e mx = — r (v), 
n ! ' 
c,{ m\n\ \ 
or is [ - = 1 (vi), 
\m n p 0 ! p x ! ... p s ! . .. (1 \) Pi (2 \)P* ... (s |)A . . ./ 
where the summation is taken for all values of p 0 , p 1} ... subject to the conditions 
(iii) and (iv). This merely expresses the fact that the sum of all the probabilities 
in the case is unity. 
Now let v - 8 ^ ) v = 8 (lPd „ _ S(vPs) 
and let a 2 - 8 ^ (P ° " Po) ^ a 2 - 8 ^^"^ 
and let <r 0 - ^ , ... cr s - ^ 
we have p s = s( — — — — ,„ ,.„ —rz ) / 8 (v) 
= 8 
p 0 lp i \...(p s )l...(l\)^C2^...(s 
m ! 
p 0 ! Pl ! ... (p, - 1) ! ...(1 !)fi(2 !)?*... (*!)*.. 
w! 
m a / (m — 1) ! \ /m" 
= — ; o 
s! V 2 , 4 ! i , 1 !...( i>s -l)!...(l!)i'i(2!)^...(s!)^- 1 ...// »l 
m (??i — l) re_s /i ! . ... 
— — i c (vii) 
s\' (n-s)l ' m n v " 
for the final summation is equivalent to (v) with p s — 1 substituted for p s , m — 1 
for m, and n — s for n, the conditions becoming 
Pa +pi +P-2 + ••• + (Ps -I) + ... =m - 1, 
Pi + 2p 2 + ... + s (p s — 1) + ... = n — s. 
