H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 369 
(iii) Samples of Four, n = i. 
Here, if a; = rp, 
_ N {l-^)^d^ / cos-i (- x )\ 
and if 
then 
TJ = cos-i (- x)jVl - x^, 
TXp 
9+1 
dx 
- qx'i'W 
q{q-l)\ x^-^Udx 
To find the first four moments about r = 0, we have to determine 
x^Udx, xUdx, 
p 
C+P 
dU 
dx 
3x^U, x^ 
dU 
dx 
Udx, 
2xU and x 
dU 
dx 
U. 
The results are most briefly expressed by using Fisher's notation p = sin a and 
remembering that 
77 
cos~i {— p) + cos~i (p) = 7T, and a = cos"'- {— p) — ^. 
Li 
For the purposes of integration we put x = — cos Q and integrate in terms of Q. 
We find 
'+'' 
J] dx = §77 (a — cos a sin a}, 
-p 
r+p 
xTJdx = 2 (sin a — a cos a}, 
-p 
+p^^ ^ 
Tra, 
sin^a ^ sin^a 
cos^a cos a 
sin^a sin*a 
Udx 
x^^ -ix^U 
dx 
dx 
3x^U = 2 
+ a 
a;2 _ 2xU = IT 
ax 
dU 
dx 
U =2 
cos^a cos^a 
sin^a ^ sin a 
cos^a cos a 
sin a sin^a 1 
a — — a. 
sin^a 
cos a 
cos^a cos^a cos a 
Hence substituting in the several values of ^J, we obtain '■ 
r = fij" = ~ {cot a + a (1 — cot^a)} (Ixxxvi), 
77" 
* Tested by formulae (Ixxx) and (Ixxxi), p. 364, which can be put in the forms 
(». - 2)(1 - p"-) dM/ _ 
P dp 
IX,' = < !(» - 2)/y- {n - 3)} 
, 1 - p^daJ 
..(xc). 
