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.

The integrated intensity, I', or 21-14 +2 cos **pw**, is thus I' =27rh,.

Of the phosphotungstic acids the most important is phosphoduodecitungstic acid, H 3 **PW**, 2040 nH 2 O, obtained in quadratic pyramids by crystallizing mixed solutions of orthophosphoric and metatungstic acids.

Of the lower qualities of Riga flax the following may be named: **PW**, Picked wrack flax.

26, celavit = ii **pw**,tie, corrupt for - ypat/ie (so Ethiopic).