92 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
H .(32), 
+ W = ......... (33). 
Clearly we require one more equation. At first sight it might seem that a fourth 
equation would come readily, from the fact that the mid-ordinate m of the frequency- 
curve is the sum of the mid-ordinates of the component probability-curves. 
This leads to 
I ^3 _ 
^y{2'7^)a■^ x/(27r)o-o 
or 
if 
v/ 
V + 
m — '\/( 27 r) mlija.. 
(34), 
But besides the disadvantage of throwing our solution back on the correctness with 
which we may have observed measurements of one size only, namely, the mean, the 
result of eliminating between (31)-(34) leads to an equation of the eighth order. To 
avoid this, it seems easier, as well as more accurate,* to take as the fourth equation 
that obtained from the sixth moment. 
Let p,ga/i® be the sixth moment of the given frequency-curve about its axis of 
symmetry, thent 
+ IScTj^Co, 
or. 
-f 
6 _ 
1 5 
(35). 
The solution of (31), (32), (33), and (35) is easy. 
Eliminating % we have, writing = v/, = v.^, 
whence 
— IX.. — iv^, 
(^f'l - «'2) = i 
{W-^ - IV = -iV/Xg - 
_ tV Me 3 _ 3 M4 
3" Mi M2^^3 Ms '^^2 
* Because our equation tLeii dejiends on all tlie observations, 
t Generally, if M,.. be the 2r moment of a probability-curve about its axis 
or, 
Mo, = (2?- - I) <r2Mo,_o, 
M.,, = (2i - I) (2;- - 3) . . . 5.3.I<rV 
