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MATHEMATICS: L. P. EISENHART 
four nets form a quatern (N,N^^\ iVf\ N12). This result constitutes a 
generalization of the theorem of permutability of transformations 
of isothermic surfaces as established by Bianchi^. In like manner we 
have the quaterns {N, Nf\ N^'\ N^) and (N^'\ N\'\ Nf\ N?). 
Moreover the six nets can be associated into the four triads N, N[^\ 
N? ; N, N','\ iVf ; N?, N?, N,, ; TVf, Nf, N,,. 
When the nets iV^J^ and N^f have been found, the functions d[ and 
d' are determined to within additive constants. Hence, if N^^^ and 
N^f are two transforms of N, there exist 00 2 nets N12, each of which forms 
a quatern with N, iV^}^ and N^2'y their determination requires two 
quadratures. 
The six corresponding points of the nets are the vertices of a complete 
quadrilateral whose four sides are generic lines of the four congruences 
which figure in the transformations. On each of these lines there are 
two focal points, each being the point of contact of the line with the 
edge of regression of one or other of the two developables as w or ?; 
varies. The four points corresponding to the variation of either variable 
lie on a Jine, and these two lines are the tangents to the parametric curves 
on the envelope of the plane of the quadrilateral; moreover, these curves 
form a net. 
Thus far we have used rectangular non-homogenous point coordinates, 
but in some cases it is advisable to make use of general homogenous 
coordinates. The four homogenous coordinates x, z, w, of a net 
satisfy an equation of the form. 
-^' + b^l + ce. (18) 
budv bu 
When two nets TV and Ni are in the relation of a transformation T, the 
tangents to the curves v = const, at corresponding points M and Mi of 
the net meet in a point Fi. Likewise the tangents at M and Mi to the 
curves u = const, meet in a point F2. It is readily seen that as v varies, 
any point on the tangent to a curve v = const, of a net moves in such a 
way that the tangent to its path Hes in the tangent plane of the net. 
Similarly for a point on the tangent to u = const., as u varies. Since 
the line lies in the tangent planes to both N and iVi, it is tangent to 
the motion of Fi as v varies, and to the motion of F2 as u varies. Hence 
Fi and F2 are the focal points of the congruence of lines of intersection 
of the planes of the nets. Following Guichard, we say that a congruence 
whose focal points He on the tangents to the curves of a net and whose 
developables correspond to the curves of the net is harmonic to the net. 
It can readily be shown that the homogenous coordinates of a net can 
