448 
PROFESSOR K. PEARSON ON .AIATHEMATICAL 
y is then found from the first of equations (viii.), or if a be the standard deviation 
= then 
Then (vi.) gives : 
y = \/{p - 3).(xiii.). 
?/.) = 
r(p -1) 
. (xiv.), 
which determines the remaining constant for the shape of the curve. 
For the position of the curve, we have for the distance from origin to mean, from 
the first of equations (vii.) ; 
= yl{p - 2) = o-v /{p - 3) 
If d be the distance from mode to mean we have : 
d = — y/p 
Further, the skewness : 
ly y 2y 
p — 2 p p(p — 2) 
Sk. = djcr 
2v/(p - 3) 
V 
Thus the solution is comjdeted. 
(XV.). 
(xvi.). 
(xvii.). 
(4.) Oa the Frequency Curve of Type VI. 
Type VI., as we have seen, corresponds to the case in which Tyj)e I. of the memoir 
has its e negative. Hence either w/ or mj is negative and the curve transferring the 
orio’in takes the form 
y — y^ {x — .(xviii.). 
Now it is possible that this curve falls under the limited range type of a frequency 
from X = 0 to X = a, but as we see that the criterion places Type VI. between two 
curves of range limited in one direction only, we expect Type VI. also to be of that 
character, and a complete solution is obtained by taking the range from a: = ct to 
X = CO ] this indeed fills iqD the gap for /Co > 1 and < oo , and (xviii.) with this range 
is seen to pass into one or other of the two transition curves 
y = Vo^Pe-^'-, 
or y = y^x-pe-^‘, 
according as we allow the first or second factor to approach a limit.! 
* The sign of jiz will determine the sign of y, or, what may be taken as the same thing, the direction 
of the axis of .r. 
t Write: y = const, x (1 - and make mi — oc, and a = cc but mi/a finite. 
Or, y = const, x (1 - and make a = 0, mi = or, and a x Hq together with nu - nii 
finite. 
