146 Expectation of Moments of Frequenc}/ Distributions 
where 
^,,„,„=1.3.5...(2s-l) 
3. 5 ...(2«-8)!i[^-i][-] + i[^- 
,, , =- 1 . 3 . 5 . . . ( 2,s' - 3) ji - + {s - 1 j 
,,0=1. 3. 5... (2.-5)1^1^ [s-2][-!+^[s-2][-J + J^V[^-2r^^ 
,,, = 1.3.0... (2^-5) {^^3 [^_2jMU+^[«_2](-sJ + ii[«-2]M+i[5-2][-^]| 
^',.„,o=1.3.5...(2s-l)i[*'-l] ^ 
„ = 1 . 3 . 5 . . . (2^ - 3) - 2]i-] + [« - + ^ - 2]^-!} 
^Vm = 1 . 3 . 5 . . . (2^ - 3) {gV - + - 2]^-^'i + tb- n-'^] 
^V,,, = 1 . 3 . 5 . . . (2« - 5) (^^Vo b - 3]t-^i + 4^.T b - 3]^-"^ + b - 3]^-" 
, , = 1 . 3 . 5 . . . (2« - 5) ! - 3][-^i + . - 3]t-«i + ^ - 3][-J 
+ M[^-3]t-' + A[«-3r^i} 
^V.,. = 1 . 3 . 5 ... (2. - 5) i3^^ [. - 3J[-] + A [5 - 3][-J + M - 3]t-^^ 
+ f* [,s-3]M + f [s-3]t-3]}y 
^ (24), 
...(25). 
0 
From (17) we find, when h > 0, 
a , 3 , 
II, II - A: i^H-k, III 
a , /3 , 
II, II — k ' II — k, III 
(h) II — k ' II — k. III 
k + m-l 
S R 
i=0 
V, , a , ^ , 
(w) H, H - A' ' n — k. 111 
r^^2^k~y'2iii —h 
(") 
r--2k+->iii 
{11} 
;=0 (or2A+2)H-/i) 
7> pll-l 
-^k, III, i ^ ii-tk--lm + h 
..(26). 
^A „ = 1.3.5...(2/. + 2m-l)0;-^,„ 
A, III, II - 'ik-%111 
Vii + li 
I ^ « 
(ii) II, n- 
k ' II - k, I 
0 
(5) In Chapter IV we shall have to deal with mcjre complicated expressions 
of type : 
()•,) ^ (>■,) O.p"''!.'-!-''! °''-2-';2-/i2 ••• a-".. Pl-l + l-2 + :.+l-k-hl-h^-...-hk,f-hi,-h2-...-h^r 
In my previously quoted paper they aie not considered as I met them for the 
first time in connection with the problems considered in the fourth Chapter of the 
present papei'. My discussion of these expressions has not so far led me to results 
which may be considered final, and I shall merely indicate the method by which 
their fundamental properties may be established. 
Putting }\ + r.,+- ... + rk = R, + /i.2 + ... + lik= H, let us replace a and by 
their values from (5) and (12). Noting that, as is well known, 
[x + y] 
[-'«] 
-i\ f,[-m-it] 
