206 
DR. W. F. SHEPPARD ON 
so that 
<R> = Vo + Vi + Vi+Vz + • • • + Vi > 
= 2/i + 2y 2 + 32/ 3 +4y 4 +...+^, 
°"2 — 2/2 + 3^/3 + 6?/ 4 + 102/ 5 +... +^Z (£— l) Vi, 
and, generally, 
V/ = Vf+(f+ L l) 2//+i + («/+2, 2) 2/ /+3 + ... + (£, l—f) Vi 
= 2 {q,f)y q \ .(25) 
9 =/ 
or, in the notation of “|Fitting,” § 4, and “ Factorial Moments,” 
oy =S"/ +1 y / ... (26) 
The cr’s can be obtained by successive summations of the y s in reverse order, I.e. 
from 2/^ to 2/0, as shown in the following diagram, in which the crosses represent 
entries in a sum- or difference-table :—- 
0 
0 
0 
0 Vi u 0 = d 0 
0 X 
0 x Vi-i 
0 x x 
0 x x Vl _ 2 
X 
X 
°5 x x y 2 
<X 4 X X 
<*3 X ?/l 
*1 
X do 
W 2 X d 4 
Ui_ 2 x 
7-1 
^1 2/o 
§ 
l-l 
X 
O'o 
(ii.) Let the d’s be the central differences of the us. Then it will be found in the 
same way that 
(a) If the u s are u 0 , u lf u 2 , ... u 2n , so that 
^0 = U n> $1 = y$ U n> S a = S 2 U n , § 2 = ..., 
