6
Similar relations exist between the elements of the other rows.
The G2: b, GZyh, Gp_ b are known by the symmetry.
The solution of Kp Gpo t° is once and for all given if both sides
of the equations are multiplied with a number so that Gh j i
The new value A G1. 2 {P-Gh 1 Gly 2) becomes the coefficient
of the solved Kp.
In the case of the matrix (8) P 3 and e.g. b being 4:
no K1
no K2
no K3
no K,
t1 t2 tz
21 8 3 1
8 24 q 3 A no.
3 9 24 8
1 3 8 21
The elements Gh h, G1; b-v Gh etc. form an autoregressive
series. Under C its special properties wiU be used.
C) The properties of the cofactors-tensor of the corrections
e*, ef=Gpa Z>e Lf"
can be derived from the place of the corrections in the scheme and
the Gpa obtained in B. The proofs will be omitted because of their
simplicity.
Property
1) The tensor is symmetrical about its main diagonal.
2) It is symmetrical about its second diagonal.
3) It is symmetrical about the line \n, \n 1 except for the
sign, which is opposite.
4) If the tensor is divided in four equal subtensors following
the lines \n, \n I, symmetry about both diagonals exists in each
separate part.
5) If in the subtensor 9=^=1, in the broken numbers
representing the absolute values of the cofactors are put under the
same denominator A, then in a column, from e1, e" to e*-1, e*
inclusive, the numerator of a cofactor is equal to the sum of the
two preceding ones. The same applies to the rows from ef, eito
ev,e*+1 inclusive.
D) The cofactors of the corrected directions become, as
ltj^= 1 (<x =(3) and o (oc^P) (see form. II, 7).
j Qpa j_fo 7,<j<p as 9 (j, and H** Gpa L«"> Lif 97^^-
The above mentioned properties 1-5 do not lose their significance
and the tensor can be written in an abridged form by folding it
along the lines of symmetry. To form the tensor Hx^n gives the
key, being equal to The sequence of the signs of H1
