MATHEMATICS: L. P. EISENHART 
173 
then X and b are apolar to f and 1/^; and the Hne bx, being the polar of 
a as to i?, has the equation 
2 ay/\/ax = 0, 
and is the tangent of F at ic. 
Dually then, ^ and 1 being the absolute points, the conic F the Feu- 
erbach circle, and the conic R a rectangular hyperbola on the four given 
orthocentric points, and having its centre c on F, if the common diame- 
ter of F and R meet R at points dd', then these points are double foci of 
circular curves of class 3 on the 6 lines; the circles with centres d and d^ 
and touching F at c are the tri tangent conies; and the two cubics touch 
F at c. 
DEFORMATIONS OF TRANSFORMATIONS OF RIBAUCOUR 
By L. p. Eisenhart 
DEPARTMENT OF MATHEMATICS. PRINCETON UNIVERSITY 
Received by the Academy, February 5, 1916 
When a system of spheres involves two parameters, their envelope 
consists in general of two sheets, say S and 2i, and the centers of the 
spheres lie upon a surface S. A correspondence between S and Si is 
established by making correspond the points of contact of the same 
sphere. In general the lines of curvature on 2 and Si do not corre- 
spond. When they do, we say that Si is in the relation of a transforma- 
tion of Ribaucour with S, and vice-versa. For the sake of brevity we 
call it a transformation R. 
It is a known property of envelopes of spheres that if the surface of 
centers S be deformed and the spheres be carried along in the deforma- 
tion, the points of contact of the spheres with their envelope in the new 
position are the same as before deformation.^ Ordinarily when 5 for a 
transformation R is deformed, the new surfaces S' and S'l are not in the 
relation of a transformation R. Bianchi^ has shown that when 5 is 
applicable to a surface of revolution, it is possible to choose spheres so 
that for every deformation of S the two sheets of the envelopes of the 
spheres shall be in the relation of a transformation R, and this is the 
only case in which S can be deformed continuously with transforma- 
tions R preserved. The only other possibiHty is that in which it is 
possible to deform the surface of centers of a transformation R in one 
way so that the sheets of the new envelope shall be in the relation of a 
transformation R. It is the purpose of this paper to determine this class 
of transformations R. 
