MATHEMATICS: L. P. EISENHART 
557 
are such that the developables of the congruence of Knes joining corre- 
sponding points of the two nets meet the surfaces on which the nets lie 
in these nets.^ 
Since Ni and A^i are applicable nets, we can obtain a cyclic system 
of circles Ci, in the manner described in the second paragraph. Hence 
each pair of solutions of (2) determines a transformation of the cyclic 
system of circles C into a cyclic system of circles Ci, such that the nets 
enveloped by the planes of the circles C and Ci are in the relation of a 
transformation T. Moreover, it can be shown that corresponding 
circles lie on a sphere. We say that two such cyclic systems are in 
relation T. 
Darboux^ has stated the results of the second paragraph in the fol- 
lowing form: If a surface S rolls over an applicable surface 5, and Q 
is a point invariably fixed to S, the isotropic generators of the null 
sphere with center Q meet the plane of contact of S and S in points 
of a circle C which generate the surfaces orthogonal to the cyclic system 
of circles C. Making use of these ideas, we give the following inter- 
pretation of the above transformations of cyclic systems : 
// N and N are applicable nets, and Ni and Ni are respective T trans- 
forms by means of (5), where 6' = Sx'^— 2^'^, the cyclic systems in ivhich 
a point sphere invariably bound to N and Ni meets the planes of contact, 
as N rolls on N and Ni on Ni, are in relation T. 
It can be shown that the two surfaces orthogonal to these respective 
cyclic systems which are generated by the points where an isotropic 
generator of the null sphere meets the plane of contact are in the rela- 
tion of a transformation of Ribaucour, that is, these surfaces are the 
sheets of the envelope of a two parameter family of spheres and the 
lines of curvature on the two sheets correspond. 
1 Guichard, Ann. Sci. Ec. norm., Paris, (Ser. 3), 20, 1903, (202). 
2 Eisenhart, these Proceedings, 3, 1917, (637). 
^ Leqons sur la theorie gdnerale des surfaces, vol. 4, 123. 
