Associated Forms.-A system of forms, such that every form appertaining to the binary form is expressible as a rational and integral function of the members of the system, is difficult to obtain.
But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R.
Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants.
A much less wise class than the 7r-computers of modern times are the pseudo-circle-squarers, or circle-squarers technically so called, that is to say, persons who, having obtained by illegitimate means a Euclidean construction for the quadrature or a finitely expressible value for 7r, insist on using faulty reasoning and defective mathematics to establish their assertions.
We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients.
Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...
Judgment is an assertion of reality, requiring comparison and ideas which render it directly expressible in words (Hobhouse, mainly following Bradley).
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.
Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.
The frequency ratios in the diatonic scale are all expressible either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.