Integral-functions Sentence Examples

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.

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.

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.

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.