MATHEMATICS: L. P. EISENHART 
453 
TRIADS OF TRANSFORMATIONS OF CONJUGATE 
SYSTEMS OF CURVES 
By Luther Pfahler Eisenhart 
DEPARTMENT OF MATHEMATICS. PRINCETON UNIVERSITY 
Communicated by E. H. Moore. June 8, 1917 
When the rectangular point coordinates x,y,z, of a surface satisfy an 
equation of the form 
a — + 0 —i W 
bubv bu bv 
the curves u = const, v = const, form a conjugate system. We assume 
that the parametric system is of this sort throughout this note, and we 
shall speak of the net of parametric curves. Equation (1) is the point 
equation of the net. 
If N is such a net, a second net iV' of coordinates x', y', z', is given by 
the quadratures 
^ = h^^y ^' = r^, (2) 
bu bu bv bv 
provided that and I' are functions of u and v subject to the conditions 
^^a(l'-h'), ^l = b{h'-l'). (3) 
ou bu 
Moreover, each pair of solutions of these equations leads by (2) to a 
net N' y which is such that the tangents at corresponding points M and 
M' to the curves of the nets are parallel. All nets parallel to N are 
obtained in this way. 
If 01 is any solution of (1), and is the function given by 
^ = h'^Jl, ^ = 1'^, (4) 
bu bu bv bv 
then the functions xi"*, y\'\ z^i \ defined by equations of the form 
00^) =oc-%x' (5) 
are the coordinates of a net iVi^\ so related to N that the lines joining 
corresponding points M and M^^^ of these nets form a congruence whose 
developables meet the surface on which these nets lie in the curves of 
the nets. We say that two nets so related geometrically are in the rela- 
tion of a transformation T. Parallel nets are in such relation. We 
