MATHEMATICS: L. P. EISENHART 175 
ba _ ^ d\/G _ a b-y/E 
bv \/e bu bu \/g' bv 
^= _ 1 ^« + ^V^ + 2mVx?sinhco, (3) 
bv -y/ E bu p2 
^ = (2J^cosha,-V£)^, ^ = (2V^sinhc.-VG)^, 
bu \ la /\ bv \ la /X 
b(a 1 bVG' OL . , doj 1 b^/E ^ , 
— = -— — — — —— smh CO, — = — = — — — —7= cosh a>. 
bu y/ E bu VXo- bv \/G bv \/\a- 
This system of equations is completely integrable provided that 
A (JL- = ^ (J- ^v^ y 
bu\\/G bv / bv\\/E bu / 
This condition characterizes surfaces with the same spherical repre- 
sentation of their lines of curvature as iso thermic surfaces. Then we 
establish the fundamental theorem : 
In order that the surface of centers of a transformation R admits a defor- 
mation into the surface of centers of a transformation R, it is necessary 
that the sheets of the envelope of spheres have the same spherical represen- 
tation of their lines of curvature as isothermic surfaces, and every surface 
of this sort admits such transformations. 
From (1) and (2) it follows that /x is a solution of the differential equa- 
tion satisfied by X, F, Z, and consequently when a transformation R 
of a surface S is known, a transformation of a surface 2 with the 
same spherical representation of its Hues of curvature as 2 is deter- 
mined by ju, the corresponding function X being given by equations an- 
alogous to (2) and the other functions follow from equations of the form 
(3). We say that the new transformation R is obtained from the given 
one by a transformation of Combescure. Hence if we determine the 
solutions of our problem for which S is an isothermic surface, the others 
may be obtained by transformations of Combescure. 
When 2 is isothermic, we may take 
Ve = Vg = e"", 
in which case the integral of the last two of equations (3) is 
= (4) 
X-fc 
where c denotes a constant of integration. 
