# Integral-functions sentence example

integral-functions

- 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.
- Algebraical division therefore has no definite meaning unless dividend and divisor are rational integral functions of some expression such as x which we regard as the root of the notation (ï¿½ 28 (iv.)), and are arranged in descending or ascending powers of x.
- The highest common factor (or common factor of highest degree) of two rational integral functions of x is therefore found in the same way as the G.C.M.
- The fundamental notion of the rational transformation is as follows: Taking u, X, Y, Z to be rational and integral functions (X, Y, Z all of the same order) of the co-ordinates (x, y, z), and u', X', Y', Z' rational and integral functions (X', Y', Z', all of the same order) of the co-ordinates (x', y', z'), we transform a given curve u=o, by the equations of x': y': z' =X: Y: Z, thereby obtaining a transformed curve u' =o, and a converse set of equations x: y : z =X': Y': Z'; viz.
- In particular if D =o, that is, if the given curve be unicursal, the transformed curve is a line, 4 is a mere linear function of 0, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 0; it is easy to see that for a given curve of the order m, these functions of 0 must be of the same order m.Advertisement
- And so if D =2, then the transformed curve is a nodal quartic; 4 can be expressed as the square root of a sextic function of 0 and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a sextic function of 0.
- It is a form of the theorem for the case D = r, that the coordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be r -0 2 .r - k 2 0 2, and writing 8 =snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.
- Algebraical division therefore has no definite meaning unless dividend and divisor are rational integral functions of some expression such as x which we regard as the root of the notation (Ã¯¿½ 28 (iv.)), and are arranged in descending or ascending powers of x.