Al. a. Tchouproff 
165 
can be expressed in the form 
= [E {fl'r, - fJ-,;) - fir.,)] ■ [E (/,., - ^,3) ifi!,^ - 
+ [E(^l\.^ - fl,^) ifX,',^ - M,,)] . [E (m',, - /U,,) (/.',,3 - )]*. 
(2) In the general case, putting 
fi,^ IX,., ... II r, = rT,j,, 
H-' I i r., ■ ■ ■ 1^' ri =n'„-,,. 
''l + ?'.,+ ...+ Ti = r, 
and agreeing to denote 
i-h + l i-li+2 i-h+n i 
t S S ... 2 bv 1 , 
we find : 
iV' N'' 
Here, i/f. = lI„-„ 
and . . may, when 2j< 'i, be written in the form 
and, when 2/ > in the form 
where 7r];U/.,y+,^ + ,..+,^^_, 
and A''^''^ denotes the sum of all products of / factors of typo /x/,jya/„ ... /x/,^ 
possible, subject to the condition thilt //j, ...hi appear in their correct order, in 
sums of not less than two terms taken from the ruunbers r;^, , ... z',-.^^, and that 
the number of summands composing Ajt is not greater than the iiumber composing 
We have in this manner: 
i-2 i-1 
+ t t 2 2 [11, +, fl,^,. + II,, +,y ^lrf_+,-r 
T n „ 
+ +.7, M-y,+.y,J „ „ a^7~ ' 
* Cf. Soper, "On the Probable Error of the Cnrrolalion Ci>elticieDt to a Second Approximation," 
p. 97 {Biometrika, Vol. ix). 
