76 Choice in the Distribution of Ohsermtions 
We shall prove that the last factor of each of these formulae is a minrimum for 
To prove this for (93) we consider the difference 
from which follows 
/^2i^4 - H-l ^ J^t^^ > 
For (94) it is at once clear that 
P-j-fA ^ /^4 -j^l 
i"2/^4-/^2-[(/^3-/^lit^2)"+/^l(^4-/^-')] jt^2M4-/Ar 
For the case of (95) we compare 
^2 t4 
and we find the difference 
\ p>o. 
and hence 
f^^i ^ H-2 ^ (A ^ 
/^2f'4 - t4 - + ^H-iH-zl^a - " l^2H-i - 14 l4- 2/^1^2/^3 + tAn-i ' 
It is thus proved for the three formulae that a distribution of observations for 
which = ij.^ = 0 gives lower values than any distribution with the same jU-g and 
^4 as the former and with /xj $ 0, /xj 5 0. 
Hence our problem is reduced to finding the /x^ and ^114 which make the following 
expressions minima : 
ar,„ = ^'(l+2a/X2 + aV4)~^, (96), 
ct;,, = ^ (1 + 2a/X2 + 0^4) — (97), 
iV ^2 
ar,, = ^(l + 2a;t.2 + aV4) (98). 
(5) Introducing /Xj = yv^ and /X4 = yz;* in (96) we get 
<. = ^(i + i^-(i + -t), 
wliich is seen to be > ^ except when y = 0. 
Q.2 
Hence the 7ninimvm value of a;,,, = ^ can only he obtained hy taking all the 
ohservations at a; = 0. 
(97) is identical with (91) for = 0. The conditions for a minimum of af,, are 
