274 Proceedings of the Royal Society of Edinburgh. [Sess. 
The expression 
becomes on this hypothesis 
[ x+c dx r 
Jx-c Jo 
, _^_cr 2 
e a2 ~ ^do- 
In this case 
( x+c dx r 
Jx-c Jo 
, - (x- (ter ) 2 _ a 2 
e 0-2 k2 da. 
A = c J 7 rk' 2, 
fx\ = v irak 
t C -1 Jd 0 7 9 
/x 2 = - + — + a k 
// 3 = ^ J irak + ty \! TraJk* + a J TrcPk 
Pi = |*(4a 4 + 12a 2 + 3) + & 2 C 2 ( 1 + 2a 2 ) + £ 
z o 
when jj.' v /ul' 2 , // 3 , etc., are the moments round the origin, which gives 
Co kJ^ • f 2 • o 
^2 = Q + 77 11 a ^ ° 
7^3 = | sjirak? 
3 & 4 7.2 2 I c4 
M4 = + *v+ 
For the form of the curve 
pr+c ( ?~ atr ) 2 0-2 
2 / = a dx I <? 0-2 * 2 o?<t 
Jcc-c ;o 
= ae 
= ae 
a2 [ X+C dx f 
js-c Jo 
'° 2 r +,! dx [V 55 “* 5 .io--aae“ a2 f V* 5 "* 2 ^ 
Jx-C Jo Jx-C J (T 
e ”* m--Wo- 
a ; 2 cr 2 
the solution of the first part is already given. 
The second is equal to 
% x-\-c 
aae- a2 Y I H' 1 ’ (ix)dx* 
Jx-c 
This method for accounting for symmetry has likewise not been successful 
in representing the statistics. 
(7) When the distribution is symmetrical around a centre it is evident 
that we have to deal with a similar integration of Professor Pearson’s form 
_ r 
and that a distribution of y = ae fc results. 
* A table of this function is given in Jahnke und Emde, Funktionentafeln , p. 135. 
