202 
DR. W. F. SHEPPARD .ON 
There are as many equations as there are coefficients a, b, c, d, ... ; and the values 
of these are thus uniquely determined. 
(iii.) The values of a, b, c, ... as found from the above equations have as their 
denominator the determinant 
0 = 
(A ; A) (A ; B) (A ; C) (A ; D) . . . 
(B ;A)(B;B)(B;C)(B;D)... 
(C ; A) (C ; B) (O ; C) (O ; D) . . . 
(D ; A) (D ; B) (D ; C) (D ; D) . . . 
There is therefore no conjugate set if this determinant is zero. The nature of the 
relations which in this case hold between the errors is considered in Appendix I., § 3. 
(iv.) Since the members of the conjugate set are l.cc. of those of the original set, 
the converse also holds. Regrouping the equations which determine the coefficients, 
it will be seen that the original set is conjugate to the conjugate set; i.e., that the 
two sets are conjugate to each other. The formulae for the members of the original 
set in terms of those of the conjugate set are 
A = (A ; A)G + (A ; B)H+(A ; C)J+..., " 
B = (B ; A) G + (B ; B) H+(B ; C)J+..., 
>.( 1 ) 
C = {C; A)G + {C ;B)H+{C; C)J+..., 
&c. 
A 
These follow from the solution of the equations in (ii.), by the ordinary properties 
of determinants; or they may be obtained more simply by determining the 
coefficients of G, H, J, ... in each case from the second of the conditions stated 
in (i.). 
(v). By means of these relations we can not only express any l.c. of the quantities of 
either set in terms of those of the other set, but we can also express any such l.c. in terms 
of particular quantities of one set and those of the conjugate set which correspond 
to the remaining quantities. We can, for instance, express any l.c. of G, H, J , K, ... 
in terms of A, B, J, K , ... by using the first two equations in (l) to determine 
G and H in terms of A, B, J, K, .... The results involve a certain determinant in 
the denominator; it is shown in Appendix I., § 4, that this is not zero if 0 is not zero, 
(vi.) Two special cases may be mentioned :— 
(a) If the errors of A, B, C, D, ... form a standard system, i.e., if the m.s.e. of 
each of the quantities is 1 and the m.p.e. of each pair of quantities is 0, the conjugate 
set is identical with the original set; and conversely. A set which is identical with 
the conjugate set will be called a self-conjugate set. 
