134 
Prohahle Errors of Frequencij Constants 
we have marked the avea, as, Heterotypic, because theoretically we need to introduce 
further terms into the denominator of our expression, i.e. to use C3. 
We now pass to a consideration of the subtypes of Type I. The equation is* 
^=^-(>-~r(>-0"". 
where and vu are roots of the quadratic 
VI- - VI (r - 2) + e - r + 1 = 0 
and r = 6(/3,-j3,-l )/(3/3i - 2/3., + 6), 
7^ 
'~4 + -l/S,(r+2)V(r+l)- 
Now nil and m^, will either be both positive or both negative if e — + 1 is 
positive, or the curve e — r + l = 0 separates the area of t/-curves or modeless 
curves from the area of modal curves (/j cui'ves) and the area of anti-modal curves 
or [/-curves. 
e — ?• -I- 1 = 0 is the biquadratic 
A {8/3, - - 12)/(4;8, - 3A) - (10^, - 12/3, - lHy/{/3, + 8)1 
This biquadratic was traced by expressing it in the form : 
A = 4(l + 2a)^(2 + a)/(2-H3a), 
and then finding /3i for a series of values of a and determining /S, from the equation 
a = (2^,~S^,-6)/(^, + S). 
Within the loop of the biquadratic all curves are ^/-curves, and the term 
" skewness " loses its essential meaning. Within this area, it will be noticed, our 
tables do not give the probable error of the skewness or of the modal divergence. 
Above this loop and up to the line 4>/32 — 3/3i = 0, we are in the range of [/"-curves 
and the skewness signifies the ratio to the standard deviation of the distance from 
mean to anti-mode. Below the loop we are in the customary Type I. area with nii 
and VI, both positive. 
Our second Diagram, B, shows what becomes of the biquadratic limiting the 
tT-shaped curves. It first meets the Type III. line, and at this point Type III. 
curves become /-curvesf and cease to have a true mode distinct from the 
asymptote value. The biquadratic then passes into the Type VI. area and 
Type VI. curves become /-curves beyond this. It never crosses, however, into 
* Pearson: Phil. Trans. Vol. 186, A, pp. 367—371. 
t Pearson : Phil. Trans. Vol. 186, A, p. 374 and Plate 9, Fig. 5. 
