REDUCTION OF ERROR BY LINEAR COMPOUNDING. 21? 
(i.) If the y s are conjugate to the us, and if 
Vr r>J =m.p.e. of y r and y s = \Js s<r , .(58) 
then (see Appendix II.) the direct (or a priori ) probability of the occurrence of the 
given set of us, if the As as given by (57) were the true U’s, is proportional to 
exp {-%2E\lr r ' t (u r -v r ) ( u s -v s )}, 
r s 
where 2 denotes summation for t = 0 , 1 , 2 , ... 1. The principle of the method of least 
t 
squares therefore leads us* to choose the e’s so as to make 
» 
22V'y,,(w r -'y r ) (u s -v s ) 
r s 
a minimum. Differentiating with regard to each of the e’s, this gives (f — 0, 1,2, ... j) 
2 1(0/) V r o,» + (l/) + i v s~~ u s) = 0.(59) 
S 
But, by (58) and (16), 
(Of) V'o,* + ( 1 /)Vu s +---+© i'l.s = m.p.e. of y s and (o f ) y 0 + (l f ) y x + ...+(2/) y t 
= m.p.e. of y s and ay,. (60) 
Denoting this, as in § 4 (iv.), by [ay], the equations (59) become (f= 0 , 1 , 2, ...j) 
2 [*/](«.-**.) = 0 . (61) 
S 
(ii.) Instead of fitting an expression of the form given by (57) to the u’s we might 
fit a corresponding expression to the ys. Since 
Vs — [ s o] ^o + [ s i] di + [^ 2 ] 4+ ••• + [ S J . (62) 
the expression to be fitted would be of the form 
z t = [ s o] e o + [ 5 i] e i + [$ 2 ] e 2 + • • • + [ s j] e j .. (63) 
* Strictly, we ought to choose the e’s so as to make B. O'exp - \P a maximum: where 
P = K - v >) ( u s ~ v s) > 
B is the direct probability of occurrence of the particular e’s denoted by e 0 , e ls e 2 , ... ej, and is therefore 
some function of these latter; and C exp - \P is the direct probability of occurrence of the particular 
values of u — v on the assumption that these values of the e’s are the correct ones, C being some function 
of these e’s. But I have assumed, as is commonly done, that the range of practically possible values of the 
e’s is so small that B and C may be treated as constants, so that we have only to consider the maximum 
value of exp - 4P. 
2 h 2 
