454 
MATHEMATICS: L. P. EISENHART 
have shown' that any transformation T oi N into a non-parallel net is 
given by equations of the form (5) . 
Let iV" be a second net parallel to N , its coordinates being given by 
(>u du bv 
and let dl be defined by 
^ = jf^, ^ = r^i, (7) 
du du bv bv 
¥ and being a pair of solutions of (3). 
Then a second transform iVf ^ has coordinates of the form 
' x^? =x-%x\ (8) 
^1 
Since the nets and iV'^ are parallel to one another, and the functions 
^1 and 6^ are solutions of the respective point equations for and N" 
in a relation analogous to (4), a transformation T of N" is given by 
=x" -% x'. (9) 
^1 
We denote by iV}^^ the net with these coordinates. By differentiating 
the expressions (9), we show that the nets and ISl^^^ are parallel. 
Moreover, it can be shown that the equations 
^1 
are consistent with the above equations, and consequently iVf ^ is a 
T transform of Hence ij a net is transformed into two nets by means 
of the same function 6, the new nets are in the relation of a transformation T. 
We say that three such nets form a triad under transformations T. It 
can be shown that the relation is entirely reciprocal in the sense that any 
two are obtainable from the third by transformations involving the 
same solution of the point equation of the third net. If in particular 
we take for di any of the coordinates of N, say 2, the nets N^^^ and Nf^ 
lie in the plane z =0. In other words, the developables of the two con- 
gruences, obtained by drawing through points of a net N lines parallel 
to the corresponding radii vectores of two nets parallel to iV meet any 
plane in two nets in the relation of a transformation T. (We postpone to 
a later time a discussion of transformations of planar nets.) 
