FOUNDATIONS OF THEORETICAL STATISTICS. 
347 
In fig. 3 are shown the contours along which the efficiency is 20, 40, GO, and 80 per cent. 
For high efficiencies these contours tend to the system of ellipses, 
8x J + 6<y' = 1 — E. 
In a similar manner, we have obtained in expression (4) the efficiency of the 
second moment in fitting Pearsonian curves. The region of validity in this case is 
shown in fig. 4 ; this region is bounded by the lines r = 3, r — —4, and by the limits 
Fig. 4. Region of validity of the second moment (standard deviation) applied in scaling of 
Pearsonian curves, showing contours of efficiency. 
())(], or m 2 , — —1) on which r 2 -fi v~ vanishes. This statistic is therefore valid for certain 
J curves, though the maximum efficiency among the J curves is about 30 per cent. 
As before, the contours are centred about the normal curve (N) and for high efficiencies 
tend to the system of concentric circles, 
1.2tc 2 +12 y 2 = 1-E, 
showing that the region of high efficiency is somewhat more restricted for the second 
moment, as compared to the first. 
The lower boundary to the efficiencies of these statistics is due merely to their probable 
errors becoming infinite, a weakness of the method of moments which has been partially 
recognised by the exclusion of the so-called heterotypic curves (r < 7). The stringency 
of the upper boundary is much more unexpected ; the probable errors of the moments do 
not here become infinite ; only the ratio of the probable errors of the moments to the 
probable error of the corresponding optimum statistics is great and tends to infinity as 
the size of the sample is increased. 
That this failure as regards location occurs when the curve makes a finite angle with 
the axis may be seen by considering the occurrence of observations near the terminus 
of the curve. 
Let 
df = kx a dx 
