K. Pearson 
197 
/x^' = KU, iJbJ = u" + v, fj,,' = 2ku(u" + 2v), /a/ = 3 + 3?)) + 
Transferring to the mean we deduce : 
lj„^ = u" + V - kHi" (xxiii), 
•/i., = KU (v - u" + 2k-u-) (xxiv), 
= 3 [ui + 3^;) + v-\ - k'-u- {27/- + 10?; + Sk-h"-} (xxv). 
fi,.2, /X3, fjb^ will be known quantities as soon as the frequency distribution is known. 
Now determine /3i = i^ilfii f^nd write X' = v/\/fM., we easily find : 
^_2^\''' + X' - ^J~/3, = 0 (xxvi). 
This cubic* gives by its real root the value of 7/ = cri — cr.,. We then easily 
deduce 
a, = iVyOa ({4 + - 3) + \')] 
y (xxvii). 
<r„ = W/x, ({4 + (4«= - 3) X'^ji - X')J 
These determine the different variabilities of the two halves. Then 
2Na, 2Nao , 
??l = , ??o = (xxviu) 
(Ti + 0-0 CTj + (To 
give the frequencies in each Gaussian curve, while 
yLl/ = «: V yU-oX' (xxix) 
fixes the position of the origin relative to the known mean value of the system. 
Thus the complete solution depends on a knowledge of the mean, and the second 
and third moment coefficients. As before the cubic is readily solved by Lill, 
Reuschle or Mehmke's mechanisms. 
The analysis is now a little more complex than in the case of the Galton- 
McAlister curve. Write e = ?;/»- = o-io-., (0-1 — o-.j)^ Then we have: 
A = /^=(e-l + 2/c=)V(l-/c'^ + 6>^ ] ^ ^ 
( XX X ) 
= {3 (1 + 3e + e-) - (2 + Zk' + 106)}/(1 - + e^J 
Thus again we see that ^ and /^^ are both functions of e only, or the skevvness is 
not independent of the kurtosisf. Whenever the skewness is zero, the kurtosis 
must also be zero or the curve be normal. 
Now consider the expression 1 — /c" + e which we will write 7, or, 
ry = -36338 + <T^<x..l{(T, - o-o)-. 
The last term is positive or 7 must be > '36338. 
* This cubic was, I believe, first given by Edgeworth. 
t The actual relation is : 
29521rj2 -f 62500/3i- - 110506/3ii; + 13468V' - 11345j3j + 1592,57; = 0, 
which, as in the Galton-McAlister case, has no obvious physical significance. 
