REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
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a representative collection of N c values of the error of C ; and so on. Thus finally 
we shall have JV = N A N B N C .■.. combinations of an error of A, an error of B, &c. 
Numbering these 1, 2, ... JV, and denoting the errors of A, of B, of C, ... by 
a, b, c, ... , the combinations will be a u by, c u ... , a 2 , b 2 , c 2 , ..., ... a N , b N , c N , ... ; and 
we shall have 
{A ; A) = (a 1 2 +a 2 2 +a 3 2 +...+a N 2 )/N, 
(A ; B) = (a l b 1 + a 2 b 2 + a 3 b 3 + ... +a lX b N )/N, 
&c. 
3. Substituting these values in 0, we find that, if there are m of the quantities 
A,B, C,..., 
iV m 0 = 
Oj 2 + Cl 2 + Cl 3 + . . . 
(i\b\ + ci 2 b 2 a 3 b 3 + ... 
a 1 c l + a 2 c 2 + a 3 c 3 + ... 
ii\ b\ + ci 2 b 2 + a 3 b 3 + ... 
b 2 +b 2 2 + b 3 +... 
byC] + b 2 c 2 + b 3 c 3 + ... 
ciyCy + a 2 c 2 + a 3 c 3 + ... &c. 
byCy + b 2 c 2 + b 3 c 3 + ... &c. 
c 2 + c 2 + c 2 + ... &c. 
= («i& 2 c 3 -.-) 3 + («i & 2 c 4 -..) 2 + {ctyb-fi...) 2 + (a 2 6 3 c 4 ...) 2 + ... 
where (ciyb 2 c 3 ...) denotes 
a x a 2 a 3 ... 
by b 2 b 3 ... 
0 c 2 c 3 ... 
Hence 0 is not = 0 unless each of the 
determinants (abc...) is = 0. This would be the case, for instance, if A were a 
constant, so that every a would be 0, or if there were a linear relation connecting 
the errors of A, B , C, .... 
4. Let be the correlation-determinant of A, B, ... P, Q, ... , and 'P that of 
A, B, ... P. 
(a) Suppose that Sf = 0. Then, by § 3 of this Appendix, each of the determinants 
(ab ... p), where a, b, ... p are the errors of A, B, ... P , is 0. But these are the 
minors of the c/s in the determinants (ab ... pq) ; and therefore these latter 
determinants are 0. Proceeding in this way, we see that the determinants 
( cib...pq ...) are all 0 ; and therefore A> — 0. 
(b) Hence, if <f> is not = 0, "P is not = 0. 
Appendix II —Frequency of Correlated Errors. 
I. Let u 0 , Uy, u 2) ... Uy and y 0 , y x , y 2 , ... y x be two conjugate sets. Denote the errors 
of the us by 6 0 , 6 X , 0 2 , ...Op, and let the resulting errors of the y s be <p u , <f>y, <p 2 , ... <p x . 
2 k 2 
