G. MORANT 
321 
We have, by (xii), 
(T- 11^) 
T- nS 
istant. 
Accordingly 
n ! V m 
where C is a constant. 
n (T-{n-l)l3r-'{T-{n+l)0r+^' 
or 
= 2n log, ( T - nj3) log, ( T - {n -l)l3}-{n + l) log, {T-(n+l)^). 
Taking the weight w,/ of Xn to be inversely as the square of its standard 
deviation, 
^ , _ 28/')t _ 8fn-i 
J n J 71+1 ./ )!— 1 
Then 
^2 _ A oj^_i _ 4 _ 4[g/„g/;,_^] 2 [g/;,+ig/;_ j 
7™ y n+i y n—i Jiijn+i J nj n—\ Jn+\Jn-i 
the square brackets denoting mean values. 
Thus approximately 
4 11 
= — I f- 
J^n fn+i fn—i 
or w'= fnfn+ifn ~i (xviii) 
J njn—i 'TjnJn+\ "r ^Jn—\Jn+\ 
To find the best value of /3 we have to make 
11' = Sw,: {2n log, {T-nS) - {n - 1) log, {T-{n - 1) /3) 
-(« + l)log,(r-(n + l)/3)-x„'j= 
a minimum, giving 
0 = 5 IV,: [2 log, (T - - {n - 1) log, (7^ - (« - 1 ) /3) 
1 
(n + l)log,(r-(j^ + l)/3)-x;] 
2?^2 (n-1)' (« + iy 
.(xix). 
.T-nj3 T-(n-l)^ T - (ji + 1) /3 > 
/3 can be found from this equation by making successive approximations, but 
the process is a lengthy one. Logarithms can be taken to the base 10 since this 
change will not affect the relative weights. 
We have next to find m when /3 is known. 
