R. A. Fisher 
515 
If now r = sin /3, 
h — tan e, 
then cos 6 = — p sin ^, 
cos 0 = - /3 Vl + /i^ sin (/S + e) = - /3 Vl + sin /3', 
and as r passes from — 1 to +1, 
/8 passes from - ^ to +~, 
6 from ^ — « to ~ + a, 
^' from — ^ + e to ^ and thence to ''^ + e, 
and ^ from ^ — « to ^ + «' and thence back to h- + 
^ Li Zi 
where sina' = p Vl +^2^ 0 oscillates in the same manner as Q, with a somewhat 
greater amplitude, and slightly in advance in respect of phase. 
The expression 
may now be reduced to 
+^1 — 0 cot ^ di 
.1 sin-^ Vl- 
p'l-^ rl- / 1 _^ _ ^ sing- sin A 
j _| sin^d) i + e Vl-sin^a'sin^/S' (1 - sin^ a' sin^/S')^/ 
^ j l-sin^a'sin^^'^'^^ j^. 
sin a' sin /3' (^/S' 
(l-sin^a'sin^/S')* 
„ fa (0) sin a' sin /S' d^' 
p^TT Trp^sma /sin e\ tto^ ,, 
= — ■/ + „ , + — (1 - cos a ) 
cos a cos^a Vcosa/ cos^a 
(1 -sin^a'siu^/SO- 
p^TT /_ sin a tan e' 
cos^ a V cos a 
but cos^a' = 1 — p'^(l + /i.^) = cos- a — sin- a tan" e, 
, , /■+^ 1 - <i cot (b dr TT tan^ a 
so that p- ■ „ , = T TT • 
J _i sin^<^ ,y'i_,.2 1 — /itana 
From this evaluation we deduce the general form 
n - 4 
(l- r') ^ dr=\n- Sir taxi^-' a (III). 
Biometrika x 66 
