MATHEMATICS: L. P. EISENHART 
639 
members of (6) and their derivatives, equation (1) is likewise the point 
equation of the net N. In view of this fact a pair of functions h and I 
satisfying (3) leads to a net N' parallel to N as well as to N' parallel to 
N. Moreover, the nets N' and N' are appHcable. This result is due 
to Peterson.2 'pj^g common point equation of the nets A^' and N' ad- 
mits the solution 
e' = + + z'^ - ?2 _ y2 _ -/2^ (7) 
This function and the corresponding function 0 given by the quadra- 
ture (4) determine transforms Ni and Ni of iV and N respectively, and 
these new nets are applicable. Hence each net parallel to one of two 
applicable nets determines a parallel to the other, and by a further 
quadrature two applicable nets which are T transforms of the original 
nets. Moreover, the function 6^ given by (7) is the only one leading to 
such a result. This result can be generalized at once to appHcable nets 
in space of any order. 
Nets are of three types with regard to applicability. Nets of the 
first type do not admit any applicable nets. Those of the second type 
admit one applicable net, whereas each net of the third type admits 
an infinity of applicable nets. We say that the latter are permanent in 
deformation, and for the sake of brevity call them permanent nets. 
Every net parallel to a permanent net is a permanent net, and each of 
the infinity of nets applicable to the one is parallel to one of the infinity 
applicable to the other by the method of Peterson. Suppose now that 
we have a permanent net N, two appHcable nets N and N, and the 
respective parallel applicable nets N\ N\ N\ By the process of the 
preceding paragraph we obtain two transforms Ni and N2 of N, in gen- 
eral distinct, such that corresponding points of N, Ni and N2 He on the 
same line, whose direction-parameters are the coordinates of N\ At 
the same time we obtain two transforms of N and two of iV. As iV 
admits an infinity of applicable nets, this process can be extended with 
the result that, in general, N and each of its deforms admit an infinity 
of transforms. We have raised the question whether in any case this, 
infinity of transforms were coincident for each of the nets so that we 
obtain a permanent net Ni, whose infinity of applicable nets arc the 
T transforms of the nets applicable to N. We refer to this question as. 
Problem A . 
Permanent nets belong to the general class of nets whose tangential 
coordinates satisfy an equation of Laplace with equal invariants. We 
have estabHshed^ the existence of transformations T of nets of this kind 
into similar nets. When in particular the given iV is a permanent net^ 
