Rit (Vísindafélag Íslendinga) - 01.06.1931, Blaðsíða 21

21
meters, a,b, etc., in the frequency distribution just now discussed
may be determined when the semi-invariants are known.
I. Let the frequency curve be represented by:
y = ab
+
1
a— b
Then we have:
. _ 1 , 1 4 = 1, 1 + = 1,1 tr
2 a2 ' b2’ 2! a3 b3’ 3! a4 b4
Dy putting P = ~ + r and q = -r we get from the
a b ab
lwo first equations:
h = P2 — 2 q and ^
p8 — 3 p q
or p3 — 3l p + ^3 — 0 (29)
This cubic furnishes us with the value of p and we are
ihen able to find q as well as a and b.
According to a familiar property of 1, fa and the higher
semi-invariants, these quantities are independent of the origin
therefore a and b may be determined, irrespective of the
origin. On the other hand Xi is dependent on this point, so
that we have 1 + xi where xi is the value of
a b
x at the lower limit, of the frequency curve in question.
K. We choose as another example the frequency curve
represented by (27a). The following relations exist between
the parameters a b, and c and the semi-invariants:
3=1.1+1
a2 ' b2 ' c2’
^-1-1.1
2 a3^b3 c3
A4
3!
1 +1
-L c*-
b4
etc.
(30)