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PROF. KARL PEARSON ON SKEW VARIATION. 
As m is positive and less than unity the area and moments of the curve are all real 
and finite. When the point (ft 2 , ft^) moves along the loop of the biquadratic towards 
the R-point (A = 1‘8, /3 X = 0), the value of 1 — rn becomes more and more nearly 
unity, and ultimately at R we have rn = 0, or the frequency curve is 
V = Vo 
a rectangle, i.e., we reach the rectangle point. If on the other hand we move towards 
infinity along the upper branch of the biquadratic loop, we find 1— m approaches the 
value 1 f/3 1 and thus ultimately becomes zero, or m — 1. Thus the limiting form of 
the frequency curve is a rectangular hyperbola, or rather the part of such hyperbola 
V = y 0 /{l+x/a) 
from the vertical asymptote x — — a to x = 0. 
But this is clearly only a theoretical limit, for it involves /3 X = /3 2 — o°,and this 
means that if /u 2 be finite, /u. 3 and /x 4 are infinite—results impossible in any actual frequency 
if the population be finite. It is clear indeed that ft 2 must be less than N, for 
obviously < N 2 /A. Again, ft 1 is < ft 2 — 1, and accordingly /3j<N —1. # But these 
limits are of small service for practical statistics, where even for small samples, say, 
N = 20, they would scarcely ever be approached.'! Thus the rectangular hyperbola can 
only be treated as a limiting form of Type VIII. far beyond the region of actual 
statistical experience-! For practical purposes the point is that m is limited to 
values between 0 and 1, or Type VIII. ranges from the rectangle to the rectangular 
hyperbola. The suggestiveness of this is that curves in the Ixj and the i.j areas, i.e., 
above and below the upper branch of the biquadratic loop, must approach these types 
as they approach the extremes of this branch. Generally a U-eurve near the biquad¬ 
ratic will be close to a curve resembling a curtailed hyperbola. 
* Mr. G. N. Watson has given me a nearer limit to ft 2 , namely, ft* < N - 2 + q—q-. But, except as 
showing that ft 2 must he finite, which is otherwise obvious, this is again of no real service. 
f The highest observed values that I know of for ft 2 and fti are those given by Duncker (‘ Biometrika,’ 
vol. VIII., p. 238). He gives 
‘ Armzahl,’ Asterina exigua N = 600 /3 2 = 33'13, fti = 1'76, 
,, Are)taster typicus N = 902 ft 2 = 128-48, ft i = 4-76. 
There are only three groups of frecpiency in each, 4, 5 and 6, and the bulk of the observations are 
concentrated in 5. The observations do not give, as he suggests, Pearson’s Type IV. and Type AT. 
curves respectively ; the i< 2 in both cases is less than unity, corresponding to Type IV. But both fall into 
the heterotypic area of Type IAL The attempt to fit with heterotypic curves would hardly be profitable 
until there was absolute certainty that the group with 4 ‘ Armzahl ’ was not the result of accident. 
\ Theoretically very high values of fti and ft 2 can easily be found, i.e., for samples of four, when the 
population sampled has, say, a correlation of 0'98; here the frequency curve for the correlation 
coefficient gives fti = 203-325 and ft 2 = 311-731, but it is the rapidly approaching zero of y 2 which leads 
to these results, 
