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.

- Since all points on any ordinate are at an equal distance from the axis of u, it is easily shown that the first moment (with regard to this axis) of a trapezette whose ordinate is u is equal to the area of a trapezette whose ordinate is **xu**; and this area can be found by the methods of the preceding sections in cases where u is an algebraical function of x.

In the case of the parabolic trapezette, for instance, **xu** is of degree 3 in x, and therefore the first moment is lh(xouo+4xlui-+x2u2).

Since PNS is a triangle of forces for the portion AP of the chain, we have wx/To=PN/NS, or yW.**Xu**/2T5, (14)

**Xu**, (23)