174 
MATHEMATICS: L. P. EISENHART 
Let X, y, z denote the cartesian coordinates of 2, expressed in terms of 
two parameters u and such that the curves u = const., v = const, 
are the lines of curvature on 2 and let the linear element of 2 be written 
ds^ = Edu^ + Gdv'^. If X, F, Z are the direction-cosines of the normal 
to 2 and pi, p2 its principal radii of normal curvature, we have the 
equations of Rodriques 
-fp =0, =0, (1) 
Ol/ ou ov Ov 
and similar equations^ in y and z. 
Darboux* has shown that the most general transformation R of 2 
is given by taking for 5 the surface whose coordinates Xo,yoyZo are given by 
Xo=x~-X. yQ = y—-Y, 2o = z — -Z, 
/U M M 
where X and /x satisfy the equations 
— +P]— - = 0, +p2--=0, (2} 
ow ow ov ov 
and by taking X//^ for the radius of the corresponding sphere. 
If Xi, yi, Zi, denote the coordinates of 2i, and Xi, Fi, Z] ; X2, F2, Z2, the 
direction-cosines of the tangents to the curves v = const., u = const, 
respectively on 2, we have relations of the form 
Xi= X ~ ^ {aXi + ^X2 + fiX), 
am 
where w is a constant, and a, jS, a are functions which are in the quadratic 
relation 
a:2 4. ^2 _|_ ^2 = 2wX(r, 
and satisfy the equations 
dju VE dM _ V^^^^ ' (3) 
dw pi Zyo pi 
da 1 d\/ 
s/g dv pi 
b«_ 1 &V£^ + ^ Vl + 2^Vx?cOsha.. 
