- The stars, as if knowing that no one was looking at them, began to disport themselves in the dark sky: now flaring up, now
**vanishing**, now trembling, they were busy whispering something gladsome and mysterious to one another. - Each of them satisfied by a common system of values; hence the equation R =o is derived on this supposition, and the
**vanishing**of R expresses the condition that the equations can be satisfied by a common system of values assigned to the variables. - For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=ï¿½.., Jz=..., and it appears that the
**vanishing**of u, v, and w implies the**vanishing**of J. - CY The proof being of general application we may state that a system of values which causes the
**vanishing**of k polynomials in k variables causes also the**vanishing**of the Jacobian, and in particular, when the forms are of the same degree, the**vanishing**also of the differential coefficients of the Jacobian in regard to each of the variables. - = = o, and have in consequence a
**vanishing**resultant.

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