Student 213 
The sum of the above series is clearly unity so that the 1st and 2nd moment 
coefficients about the zero end of the series ai'e given by two series of which the 
?'th terms are 
rS[qiq.,-qr{l - q^+i) ••• (1 - <ln)\ '^id r'S [q.q., . . . q^{l - q^^,) ... (1 - q,,)\ 
respectively. 
These series may be summed by rearranging them in the ascending order of 
the q products thus : 
S\q,{l-q.Ml-qs)...(l-qn)]=S(q,)-2S(q,q.^+...+{-iy-^r.S(q,q,..^ 
28 {qMT^-<h)a-'h) • • • (1 = 2S{q,q,) + ...+(- 1 )'-'>■(>■- l)S{q,q, . . . q,)+... 
tS {q,q,...qt{l- qt+i ) • • • ( 1 - '/u)} = {q, q., . . . q,) + . .. 
rS {q,qo ... 2, (1 - qr+i) ... (1 - qn)] = r.8{q,q.,... q,) + .... 
Adding these we get on the left v-^ and on the right 8{qi) + a number of terms 
of the form r (1 — l)*""^ ^{qiq^ ... q,) which accordingly vanish and we get 
v. = S{q,). 
In a similar manner it can be shown that 
V, = S{q,) + 28{q,q.^, 
and other moment coefficients about zero can be found in the same way, but we 
are not here concerned with them*. 
If q, q- are the mean values of q and q\ obviously 
v^ = 8(q,) = )iq (1)^ 
and i^,= S(q,) + 28{q,q.^ = 8 (q,) + [8{q,}}-^- S{qr) 
= nq + irq' — nq^ (2), 
= nq + ri'q'-nq- - na.f (3), 
. ■. ytto = nq — nq^ — na,f 
= -5- y) • (i). 
* The moment coefficients are : 
A'3 = np<i (p-q)-^n{p- '<i) qP-i + 2rt 
/i4 = nfq {l + -d{n-2)pq}-n{l + Q{n-&)j>q]qixo + 12k {p - q) ^/i., - 6w,;^4 + 3h2,^./, 
where 3;U2 etc. are the moment coefficients of the q distribution and p = \-q. 
