286 
On the Systematic Fitting of Carves 
Chordal area of Zx^ = {z^ -\- z-^ + + ... + Zn) 0^ 
+ + ^2 + . . . + Zn) I' 
+ (^2 + ^3+ ... +^„)2* 
+ {Zn-i + Zn) (n - ly 
= ^„(P + 2« + 3''+ ... +n«) 
+ Zn-, {V + 2'+S'+ ... + {n-l Y) 
+ Zn-2{l' + 2' + 'i'+ ... + in-2y) 
+ 
+ 
+ z,(V + 2') + z,{V) (xii). 
Now 1'+ 2'" + 3*+ ... + }f can be summed by a Bernoulli's numbers series, i.e. 
l« + 2« + 3«+ ... +n' = , +^n' + --'n'-' 
s + 1 |2 
s{s-l){s-2) sis-l){s-2)(s-3)(s-4) 
the series ending with a constant or n, according as s is odd or even. 
Now we may write on the right-hand side n + ^ — ^ for n, and we find accord- 
ingly that 
...(xiii). 
_3_ 
64 
2(1 +2 + 3 +. 
.. + n ) 
= («+l)^- 
1 
¥> 
3 (12 + 2^ -f- 32 4- ., 
. . + 
= + 1)^ - 
i(« + i)> 
4(13+23 + 33+ 
.. +n^) 
= (« + - 
Hn + hy + 
1 
1 6 
5(1^ + 2^-1-3^+., 
..+n*) 
= (n + if - 
7 
48 
(n + i)- 
6(l= + 25 + 3=+ ., 
. . + iv') 
= {n + hY- 
1 (n + + 
7 
16 
(n + i)= 
In these we can write n, n — \, n — 2, n — etc., successively for li, but z^ (r + |-y is 
n 
the sth moment of z^ about one end of the range, and Szr{r + ^y is the sth 
0 
moment of the system of grouped frequencies about one end of the range. Let us 
call this Nvg. We can now rewrite equations (xi) in terras of the ^-''s. We first 
note, however, that when s — Q: 
fj-i = y j Zdx = -y- (chordal area) by (x) 
= {^Z, + Z,+Z,+ ... + hZn)/jS^ 
= {(n + ^)zn + (n - 1 + ^) Zn-, + (n - 2 + i) Zn-2 + etc.}/iV 
