For this reason the umbrae -a 2, a l are said to be cogredient to 5 1 and x 2.
We frequently meet with cogredient and contragedient quantities, and we have in general the following definitions:-(i) " If two equally numerous sets of quantities x, y, z,...
By the same scheme of linear substitution the two sets are said to be cogredient quantities."
(I.) Introduce now new umbrae dl, d 2 and recall that +d 2 -d 1 are cogredient with x, and x 2.
We may in any relation substitute for any pair of quantities any other cogredient pair so that writing -}-d 2, -d l for x 1 and x 2, and noting that gx then becomes (gd), the above-written identity bceomes (ad)(bc)+(bd)(ca)+(cd)(ab) = 0.
The repetition of the process brought the same results.
Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),...
In both cases ddl and dal are cogredient with xl and x 2; for, in the case of direct substitution, dxi = cost dX i - sin 00-(2, ad2 =sin B dX i +cos O dX 2, and for skew substitution dai = cos B dX i +sin 0d2, c-&-- 2 n d =sin -coseax2.
Ai by substituting for y 1, y 2 the cogredient quantities b2,-b1, and multiplying by by-k.