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MATHEMATICS: C. A. FISCHER 
by the solution of a completely integrable system of partial differential 
equations of the first order a family of parallel nets of a particular type 
are obtained, each of which determines a T transform A^i, which also is 
a permanent net. All of these transformations are now shown to give 
a solution of Problem A. 
In the transformations just referred to we did not consider permanent 
nets for which the curves in one family are represented on the Gauss 
sphere by one system of the imaginary generators. Drach* solved the 
problem of the deformation of nets of this kind. We show how in two 
ways these nets can be transformed into nets of the same kind as a 
solution of Problem A. 
The third type of permanent nets are those whose two families of 
curves are represented on the sphere by its isotropic generators. These 
curves are the minimal lines on a minimal surface. There are no trans- 
formations of nets of this kind into similar nets furnishing a solution 
of Problem A. 
lEisenhart, Trans. Amer. Math. Soc, New York, 18, 1917, (97-124). 
^Peterson, Ueber Curven und Flachen, Moskau and Leipzig, 1868, (106). 
sEisenhart, Rend. Circ. Mat., Palermo, 39, 1915, (153-176). 
^Drach, Ann. Fac. Sci. Toulouse, (Ser. 2), 10, 1908, (125-164). 
ON BILINEAR AND N-LINEAR FUNCTIONALS 
By Charles Albert Fischer 
DEPARTMENT OF MATHEMATICS. COLUMBIA UNIVERSITY 
Communicated by E. H. Moore, October 1. 1917 
It has been proved by Riesz^-^ that if a linear functional A{f{s)) 
is continuous with zeroth order, there is a unique regular function 
a {s) which satisfies the equation 
A if) =ff{s)da(s), 
and the variation of a is the least upper bound of the expression 
\A(f) I /mf, where mf is the maximum of \f{s) |. From this theorem 
Frechet]^ has proved that if U (/ {s),g (/) ) is bilinear, that is linear 
in each argument, there is a function u {s, t) which is regular in t and 
satisfies the equation 
U{f,g)=fff{s)g{t)d2u{s,t), (1) 
and by modifying the definition of the variation of a function of two 
variables, he has proved that the variation of u {s,t) is the least upper 
bound of I U (f, g) \ /mfmg. 
