R A. Fisher 
513 
6. The direct process of integration by parts applied to such expressions as 
71 - 4 
(1 - r-) 
-1 
dr and j (1 — r^) 
8^* 
when 71 is even, merely introduces the sums and differences of the terms ^ 
at the extremes, where r is — 1 or + 1, with coefficients which are, in any 
particular case, easily calculable. 
Thus, n being 6, 
8= 
dr' 2 
(1 - r^) 
^ ^1 
dr* 2 
+ 
2r 
3-^ ^2 
+1 
-1 
dr- 2 
+1 
-1 
P 
= 2 X the sum of the extreme values of - . , ^ — 3 cot 6 + "86 cot" 6) 
sin'* 0 
— 2 X the difference of the extreme values of -.-^-^ (1 — 6 cot ^). 
sin^ a 
If p = sin a, so that the extreme values of 6 are — a and ^ + a, the sums and 
differences may readily be expressed in terms of a, and the first few may here be 
tabulated : the table has been carried back as far as is necessary for the calculation 
of the fourth moment. 
sin^ e f7 + 2(92 
sin 6 
P 
2 
+ 1 
-3(9 cot^-^-^ cot2^| 
sin 6 
sin^ 6 
sin^ a 
sin* a 
(l-^cot(9) 
(^-3 cot ^ + .3^ cot2^) 
(4 - 9(9 cot ^+ 15 cot2 ^ - 15(9 cot^ ^) 
77 cot a (1 +a tan a) 
-4 
rr tan a 
2 tan^ a (1 H-a tan a) 
77 tan''* a (1 + 3 tan^ a) 
2 tan* a (4 + 9a tan a + 1 5 tan''^ a + 1 5a tan^ a) 
77 cot^ 6 
diflereuce 
(a + 3 tan a +3 a tan^ a) 
cot a |2a - 2 tan a + + a'^^ tan t 
77a 
2a tan a 
77 tan^ a 
2 tan^ a (a + 3 tan a + 3a tan^ a) 
77 tan* a (9 tan a + 15 tan^ a) 
There are here two natural series, which appear alternately as sums and 
differences ; the simpler, which may be expressed in the form 
TT 
sin^ a 
cos add 
