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MATHEMATICS: L. P. EISENHART 
When we take c = 0 in (4), and substitute this value of w in (3), 
the latter reduced to the equations of transformations Dm of the iso- 
thermic surface 2 into a new isothermic surface Si.^ In this case the 
linear element of the sheet of the new envelope arising from the de- 
formation of the surface of centers S of the transformation Dm is 
and the linear element of its spherical representation is 
J^i2 = (f! + e-A' du' + (- - ^ e-*')' dv\ (5) 
The other sheet of the envelope, namely S'l, is merely a point, say 0. 
Conversely as we have shown elsewhere^ also, a Dm is the only transfor- 
mation R whose surface of centers can be deformed so that one of the 
sheets of the new envelope is a point. 
From our fundamental theorem it follows that (5) is the spherical 
representation of the lines of curvature of two isothermic surfaces 2 
and S', themselves Christoffel transforms of one another. Their respec- 
tive linear elements are 
When we apply to 2 and 2' transformations D^ determined by the func- 
tion /X which is involved in the transformation from 2' into the point 0, 
we get two new isothermic surfaces 2i and 2'i, whose linear elements are 
respectively 
d's^"- = X^^-^^ {du^^dif), d's,"" = — - (du^ + diP), 
The surface 2/ is the one which Bianchi^ defined in a purely intrinsic 
manner and called the transform of 2 by the transformation deter- 
mined by the D^ of 2 into 2i. 
Let 6*0 be the surface of centers of the spheres enveloped by 2' and 0. 
Corresponding normals to 2', 2, 2' are parallel, and in like manner the 
normals to 2i and 2'i are parallel to the lines joining O to the points on 
5*0. Hence the permanent conjugate system on 6*0, that is the system 
corresponding to the lines of curvature on 2, project upon the unit 
sphere with center at O into the orthogonal curves representing the lines 
of curvature on 2'i. 
