PROF. KARL PEARSON ON SKEW VARIATION. 
439 
then a is determined from 
a = ±er( 2- rn) a/ ~ —— , 
V l —m 
the sign being determinable from the observed value of ju 3 and y 0 from 
N(l-m) 
?/o = —- 
a 
N I — m / 1-772 
a 2 — m \ 3 — m 
and the placing of the frequency curve on the observations by 
//] = a (L — m)/( 2 — m). 
If, however, we find 108,—12&—18 and 3^,— 2/5, +6, we have 
giving 
o o 
10/3,-12/3,-18 = + 
3,8, —2/3,+ 6 = - 
24m (m — 2) :: 
(l — m) (5—m) (4—m) 2 ’ 
24 (m —2) 3 
(1 — m) ( 5 — m) ( 4 — m ) 2 ’ 
2 (5/3 3 —6/3, —9) 
3/3, -2/3, + 6 
and thus since m is to be positive, the point (/3„ 82 ) must be above the line 
58, — 68 i —9 = 0. The line 2/3o—38i — 6 = 0 does not meet 5/3, — 6/3i — 9 = 0 in the 
positive quadrant, so that a point below both these lines does not exist in real 
frequency. Clearly 
1-m = (882-981- 12)/(38 i-28 2 + 6), 
3- m = (48,-38 1 )/(38 1 -28, + 6), 
4— m = 2(82 + 3), 
and thus if these values be substituted in 81 as given above, we reach 
81 (82+ 3) 2 (88,-98i-12) = (48 2 -38.) (IO82-I281-18) 2 
the equation to the biquadratic, proving that the point associated with the above 
frequency curve lies on the biquadratic. 
Again 1 —m will always be positive, or m less than unity. For the upper branch 
of the loop of the biquadratic lies below its asymptote, or 8 ( 3 2 — 98i— 12-f = 0, and 
accordingly below the line 88 , — 98i —12 = 0 ; thus the numerator of 1 —m is always 
positive. So also is the denominator, for the upper branch always lies above the line 
28 2 -38i-6 = 0 .* 
* In fact the R-line (5 82 - 681 -9 = 0) the parallel to the asymptote (8/3o - 9/2, -12 = 0), the limiting 
frequency line (/+- 81 -I = 0 ), and the Type III. line (2f3- 2 - 3/3 - 6 = 0 ) meet in the point 82 = - 3, 
81 = - 4 of the negative quadrant and the upper branch of the loop lies in the angle between the first two 
and in the positive quadrant. 
3 O 2 
