By Stitdknt 
11 
Hence to find the area of the curve between any limits we must find 
. 'i^^ ... etc. X f c.s"-^ e,ie 
n - S )i — D J 
cos""-'' f> sin 6 
— 2 
= , . ; . . . et '. u cos" d(W 
n — '4 II — -i [a — z] 
= . ... etc. cos"-^ Odd H , . . . . etc. [cos"-' 6 siii 6\ 
n - 0 n — 7 .' n — H h - o 
and by continuing the process the integral may be evaluated. 
For example, if we wish to find the area between 0 and 6 fur n = 8 we have 
area = ~ . . - . cos" 6/ dv 
5 S I TT./,, 
4 2 r« ,,,^142 
O TT 
cos^ ^r/^ f - . . - cos"' 6 sin 
O ^5 TT 
= — h - cos 6 sm 0 + - . - COS'' 0 sin + - . , . cos' sin d, 
TT TT O TT O 4 TT 
and it will be noticed that for h = 10 we shall merely have to add to this same 
1 G 4 2 _ /, . ^ 
expression the term ^ . . ■ :j • ^ cos' 0 sin 6. 
The tables at the end of the paper give the area between — oc and z 
or ^ = — ^ and 6 = tan^' 
This is the same as '5 + the area between ^ = 0, and ^ = tan~^ 2', and as the 
whole area of the curve is equal to 1, the tables give the probability that the 
mean of the sample does not differ by more than z times the standard deviation 
of the sample from the mean of the population. 
The whole area of the curve is equal to 
J) 2 H 4 /' * 
~ ^, . ... etc. X cos"—- ddd, 
11 — 6 n — r) J _n 
and since all the parts between the limits vanish at both limits this reduces to 1. 
Similarly the second moment coefficient is equal to 
etc. X cos""- ^ tan- Odd 
n — :5 n — 5 
?( — 2 ?i — 4 
91-3 ■ 'H-5 
etc. X I (cos"-^ 9 - cos"-2 0) d0 
