MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 379 
Thus this expression is again critical for the class of curve with which we are 
dealing. We may say that a skew frequency curve will have limited range, range 
limited in one direction only, or unlimited range according as 
Zp (Bg? — py) $F 3prs” 
is greater than, equal to or less than zero. Thus the calculation of this expression is 
the first step towards the classification of a frequency curve given by observation. 
(i1.) It is noteworthy that the values we have obtained for 7, z,a,v and y, will be 
real and possible if 7 > 1. On the other hand we have required in our work that r 
should be > 3. I propose now to return to this point. So long as 7 > 1 the values 
of both pw’, and py will be finite, but the values of p’; and py’, and consequently of ps 
and p, will be infinite if r be < 3. That is to say, the third and fourth moments 
of the curve about the centroid vertical become infinite. This is quite conceivable 
from the geometrical standpoint, and various interesting questions, of purely 
theoretical value however, arise according as 7 > 1 and < 2, 2.¢., 4, and ps are both 
infinite, or 7 > 2 and < 3, 2.e., w, alone is infinite. The solution we have given fails 
in these cases. We should obtain, however, finite relations between the four constants 
of the equation to the curve by taking the first and second moments am”, and ap’, 
round the axis of x; we find in this case 
pe = ty 
Ly “costr29 o-2 6, 
—7/2 
Op» = AY 0° ee cos*"t#9 e—8 8, 
—1r/2 
or, 
1 = dye *" G (2r + 2, 2v)/G(r, v), 
p'n = dye G (Br -b 4, 37)/G (x, »). 
These results together with 
are theoretically sufficient to determine the four constants 7, v, yy and a. Practically 
they would hardly be of service without very elaborate tables of the G functions. 
As a matter of fact, we are very unlikely in dealing with actual statistics to meet 
with cases in which pz and p, become infinite, because neither the range of observa- 
tions, nor the size of the groups observed at great distances from the origin can be 
infinite. With finite values of pz and py, it is, however, easy to see that we always 
obtain from our solution on page 377 a value of 7>8, so that the solution is self- 
consistent. 
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