By Student 
353 
Hence the second moment-coefficient about the mean 
fjL., = m + v;i- — vi- = m. 
By similar* methods the moment-coefficients up to fi^ were obtained, as 
follows : 
ni. 
m. 
f^i = 
m. 
3«i^ -H m. 
fJ'^ = 
lOrti'-^ + m. 
] 5»i^ -1- 25«i- -I- 
Hence /Si = ^ = — , 
and ^, = i^ = 3-H-. 
It will be observed that the limit to which tliis distribution approaches as m 
becomes infinite is the normal curve with its ^i, ^83, ySj, etc., all equal to 0, and 
^, = 3, /34 = 15, etc. 
Further, any binomial {p -i- (/)" can be put into the form (p -f- q')"' '', and 
if q be small and nq not large it approaches the distribution just given. 
Thus if 1000 (y-^ + j^o)'"" be expanded the greatest difference between any 
of its terms and the corresponding term of 1000 e~'' il + o -\-'^^+ ... + — + ...) 
* The evaluation of the moments about the puint O will be found to depend on the expansion of r" 
in the form 
(r-1)! (r-1)! (r-1)! 1) !) 
(,. - „ - 2) ! + "1 (r - „ _ 1) ! + *2 _ „j ; + • • • + _ 1) 1^ 
+ 7 — -,. + ...+ 
(((•-n-2) \ (r n-l)\ [r-n)\ (r 
Then if we form the series for )t + l from this it will be found that the following relations hold 
between a^, a„, a.^ etc. and the corresponding coefficients for » + !, ^j, A.,, A.^ etc. 
A„ =((.,+ (11 - 1) «!, 
^l,, = "p+(«-p + l)Op-i. 
From these equations we can write down any number of moments about the point O in turn, and 
from these may be found the moments about the mean by the ordinary formulae. 
The moments may also be deduced from the point binomial (p + j)"^ when q is small and n large 
and nq=m, i.e. p = l, q—0, nq = in. We have 
fj,^ = nq = m, 
fi2 =npq=in, 
Ms =npq(p- q) = m, 
/jL^ =np)q {1 + 3 (« - 2)^5} =7H (1 + 3Hi) = 3»i- + ?7i. 
