It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products.
If, however, an amount of energy a is taken up in separating atoms, the ratio is expressible as C p /C„= (5+a)/(3-Fa), which is obviously smaller than 5/3, and decreases with increasing values of a.
Also, as the Cartesian geometry shows, all the relations between points are expressible in terms of geometric quantities.
P. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are expressible as rational and integral functions.
Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.