430 
PROF. KARL PEARSON ON SKEW VARIATION. 
causes is no longer indefinitely large, and the contributions of these causes are no 
longer independent but correlated.* 
Since and /3 2 are by nature positive we can represent all possible values of /3 1 on a 
chart in which (3 X and j3 2 are the co-ordinates of a point in the positive quadrant. But a 
little consideration shows that (3 2 must be greater than /3 l , thus one-half the area of the 
quadrant, that above the line /3 2 — /3 X is removed from the field of possible occurrences. 
Further, there is a limit to the application of the series of curves discussed when /3, 
gets large, for the high moments of two of the types of curves, i.e., Types IY. and VI., 
or 
v tan 1 x ja 
V 
■] (x—a) q ' z 
and y = y t) - -- 
x 
1 1 
become infinite when the order of the moment is greater than r, or the probable error 
of the fourth moment would become indefinitely large for r = 7, i.e., we are practically 
limited by the line 8/3 a —15^ — 36 = 0. The first four moments of the curve remain 
finite, but from the fifth onwards they can become infinite, the lines corresponding to 
these, however, lying outside the above line.! For curves corresponding to points 
below this line it is fitting to take as differential equation 
1 dy _ _ b + x _ 
y dx c 0 + cyx + c 2 x 2 + cpt? ’ 
or a slightly more general form which is related to the higher hypergeometrical 
F (a, (3, y, 6, e, l) as the present series of curves to the simple hypergeometrical 
F (a, (3, y, l). The whole theory of curves of the above type has been worked out for 
some time past, but has remained unpublished, for we failed to find any definitely 
homogeneous data by which it could be effectively illustrated, and for this reason 
heterotypic curves have for the time being been left in abeyance. We may, however, 
notice the following point. If we take our generalised hypergeometrical to be 
1 , a ■ ■ y , ( a + l)(/3+l)(y+l)a./3. y , 
0.e.£ (0+l)(e+l)(f+l )6>.»,.f 
Then 
— 2/u + 2/i + 2A+ ••• • 
y : c+\ _ (a + cr) ((3-\-x) (y + a?) 
y x (0 + cc) (e + a?) (£+as) 
and this will correspond to the ordinary form if f = 0, i.e., F (q, (3, y, 6, e, l). 
* Just as values of the binomial (p + q) n with negative n andj?>l very often give good fits to frequency 
distributions, so we have recently found that hypergeometricals F (a, /3, y, 1) with imaginary a. and /? are 
of fairly common occurrence in frequency distributions, and when applied to individual samples from real 
hypergeometrical populations may give better fits than the theoretical series, i.e., in card drawings, 
t See Rhind, ‘ Biometrika,’ vol, VII., p. 133. 
