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),...

Ai by substituting for y 1, y 2 the **cogredient** quantities b2,-b1, and multiplying by by-k.

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.