By Student 
The second moment about the end of the range is 
The third moment about the end of the range is equal to 
I a+l n+ \ 
J- n—2 ^ n—i J- n—i 
= a" X the mean. 
The fourth moment about the end of the range is equal to 
(»-l)(M+l) ^ 
7 = s cr • 
Dc 
If we write the distance of the mean from the end of the range -7— and the 
^ ^Jn 
moments about the end of the range Vi, Vo, etc. 
^, Da- n. - 1 Ba^ rr - 1 
tlien Vi = -r^, v., = cr-, v-, = ~r~ , v. = a* 
From this we get the moments about the mean 
11 ' 
At3 = ^ {nB - 3 (h - 1) i) + W-^] = ^ (2/)^ - %i + 31, 
n sin ^ ' n tjn ^ ' 
= {,^2 _ 1 _ 4i)2,i + G (/i - 1 ) i»- - = ~ {/i^ - 1 - D-^ - 2n + 6)}. 
It is of interest to find out what these become when ii is large. 
In order to do this we must find out what is the value of D. 
Now Wallis's expression for tt derived from the infinite product value of sin x is 
If we assume a quantity d ^= o„ + — + etc.j which we may add to the 2n + 1 
in order to make the expression approximate more rapidly to the truth, it is easy 
to show that ^ = — ^ + ~ etc. and we get 
2 1 on 
IT ] 1 \ 2-. 4-. e--' ... {-iny 
•2 \ 2 r-.3-.5-...(2'H-l)-^' 
3 1 
From this we find that whether n be even or odd Z)^ approximates to ^ ~ 9 + g^^ 
when n is large. 
* This expression will be foixnd to give a much closer approximation to tt than Wallis's, 
Biometrika vi 2 
