344 
MR. R. A. FISHER ON THE MATHEMATICAL 
the lines AC and AC' represent the limits along which the area between the curve and 
a vertical ordinate tends to infinity, and on which m u or m 2 , takes the value — .1 ; the 
line CC' represents the limit at which unbounded curves enclose an infinite area with 
the horizontal axis ; at this limit r — —1. 
The symmetrical curves of Type II. 
a i 
extend from the point N, representing the normal curve, at which r is infinite, through 
the point P at which r = —4, and the curve is a parabola , to the point B (r = —2), 
where the curve takes the form of a rectangle ; from this point the curves are U-shaped, 
and at A, when the arms of U are hyperbolic, we have the limiting curve of this type, 
which is the discontinuous distribution of equal or unequal dichotomy (r = 0). 
The unsymmetrical curves of Type I. are divided by Pearson into three classes 
according as the terminal ordinate is infinite at neither end, at one end (J curves), or 
at both ends (U curves) ; the dividing lines are C'BD and CBD', along which one of 
the terminal ordinates are finite (m x , or m,, = 0) ; at the point B, as we have seen, both 
terminal ordinates are finite. 
The same line of division divides the curves of Type III., 
<1f cc x p e~ x dx, 
at the point E (p — 0), representing a simple exponential curve ; the J curves of Type III. 
extend to F (p = —1), at which point the integral ceases to converge. In curves of 
Type Ilf., r is infinite ; v is also infinite, but one of the quantities m x and m, is finite, 
or zero (— p) ; as p tends to infinity we approach the normal curve 
df oc e~ ix ' dx. 
Type VI., like Type III., consists of curves bounded only at one end ; here r is 
positive, and both m 1 and m., are finite or zero. For the J curves of Type VI. both 
m i and m, are negative, but for the remainder of these curves they are of opposite sign, 
the negative index being the greater by at least unity in order that the representative 
point may fall above CC' (r — —1). 
Type V. is here represented by a parabola separating the regions of Types IV. and VI.; 
the typical equation of this type of curve is 
_ r+3 _ 1 
df cc x 2 e x dx. 
As r tends to infinity the curve tends to the normal form ; the integral does not 
become divergent until = 1, or r — — 1. On curves of Type V., then, r is finite 
or zero, but v is infinite. 
