158 Expectation of Moments of Frequency Distributions 
V 
Equations (15), (16), (17) and (18) hold for all laws of distribution of the 
variable A". If the law fulfils the conditions 
^,,.+, = 0 for 1 = 0, 1, 2, ... r, 
then /u-o,.+j,(_v) = 0, for all the coefficients then vanish. As regards the 
coefficients Tor^,—h, they take the form: 
T.r,r-h = 1.3.5... (2r -1)1 r^-(h+in 2^+1 /^,r-h-i 
! = l 
X z 
where the last summation extends to all positive integer values _yi,j2, •••jf, satis- 
fying the condition: + ... +jf = i, and to all integer values of li^, h.^, ... hj, 
satisfying the conditions : 
2 ^hi< h.,< ... < hf, 
lt,ji + h-J-i + • . . + hfjf — h + i. 
If the law of distribution of values of X also satisttes the conditions : 
//..,,• =1.3.5... (2i - 1)^./ for i = 1, 2, 3, ... r, 
then (cf Introduction (5), (8) and (16)) 
T^,.^,_n = 1 . 3 . 5 ... (2/- - 1) i ,■[-'''+"! 2''+' yu/ x 
i = \ 
^ [1 . 3 . 5 . . . ( - 1 )]>' [1 ■ 3 ■ 5 . . . ( 2/i, - 1 )-\h . . . [1 . .-3 . 5 . . . (2/t^ - 1 )]^/ 
i:!j.!---i/![(2/0!p'[(2A.)!p^...[(2^,)!]V 
= 1.3.5... (2v - 1) yu/ of,.,,._A, 
and 
(•-I J ; 
/^3,-,(-V)=l -3.5 ... (2r- l)/i/ 1 -j^. :£ (- ly a,.,r-i+h (3,-i+h,h 
= 1.3. 5 ... (2r - 1)M/ ^, = 1.3.5... (2r- 1)^./,(a.). 
Thus in the case when the law of disti-ibution of the values of X for i ■^r 
gives values of m answering the Gauss-Laplace law, then the values X(^y) follow 
a law of distribution giving for for i ^ the same values as the law of Gauss. 
Consequently if the law of distribution of values of X is Gaussian, then the law of 
distribution of values of X is Gaussian also for all values of A^. 
