MATHEMATICS: L. P. EISENHART 455 
If 0o is any solution of (1), a solution of the point equation of N[^^ 
is given by 
^12 = ^2-5^2. (11) 
This function and the net N^^^ parallel to N^^^ determine a transfor- 
mation of the latter; moreover, the congruence of the transformation 
consists of the joins of corresponding points on iVi^^ and iV^. We call 
the transform and its point coordinates Jvi-, z^. The solution 
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of the point equation of N[^^ corresponding to 6^2 is of the form 
dn - ^2 - 5 (12) 
and consequently we have 
= 4') - ^f] - ^f'l (13) 
The function $2 can be used to determine with the nets and N'' 
two transforms of N, namely N[^^ and iVf'*, whose coordinates are of 
the respective forms 
= X - x', xT = x-% x". (14) 
62 O2 
Corresponding points of the nets N, Nf\ and iVf^ He on a line, and 
is a transform of Nf^ by means of the function 
^2-5 Ol (15) 
Likewise, corresponding points of the nets N[^\ Nf\ N12 lie on a line, 
and Ni2 is a transform of TVf ^ by means of the function 
On - (16) 
^1 
By means of (11) and (12) we show that the expressions (15) and (16) 
are equal, and consequently the nets Nf \ Nf \ fo™ a triad. 
Equation (13) is reducible to 
^12 = ^ + [{e[ $2 - 626 ;)x' + {e\e^ - e[e2)A/{e[e2 - ^X) (17) 
From the symmetry of this expression we see that N and N^2 ^-re trans- 
forms of N^^^ and Nf^ in an analogous manner. We say that the 
