R. A. Fisher 
511 
which, on substituting cos 6 for — pr, may be expressed in terms of a Legendre 
function in the form 
jt - 4 
(icosec^)"-iQ„-2(*cot^).(l-r-) dr (II). 
r cfe e 
° j 0 cosh 2^ + cos ^ sin^' 
p dz 1 / d y-^ 
J 0 (cosh z + cos 6')»-i ~\n-2 [sm Odd J 
so that 
(cosh ^ + cos |n-2Vsin^9^/ sin^' 
and since this is a function of pr only, we may express the frequency distribution 
by the convenient expression 
n — 4 
3"-^ / e 
Professor Pearson has shown that this last result can be obtained directly 
from Sheppard's theorem* that 
making the substitutions 
1 n 
1 71 
R tirp 
which give R = pr 
and cos~^ { — R) = 6, 
we obtain 
sm( 
and hence differentiating {n — 2) times with respect to r, the required expression 
is obtained. 
5. The form which we have now obtained may be applied without difficulty 
to all small even values of n, and in such cases is peculiarly suitable for the 
calculation of moments. 
When n = 2 the ordinate of the curve, with abscissa is 
(1 - f') sind' 
which becomes hyperbolic in the neighbourhoods of — 1 and + 1. The value 
■ * Phil. Trans. Vol. 192, A, p. 141. 
