MATHEMATICS; L. P. EISENHART 
63 
faces of the same kind, say Co and Co', such that the congruences of lines 
MMq and MMq are normal. Furthermore, the spheres of radium t with 
their centers on C are enveloped hy two surfaces, each of which is orthogonal 
to one of the congruences. These orthogonal surfaces are surfaces as 
defined by Demoulin,^ who showed that they are characterized by the 
property that their fundamental coefficients E, G, D, D" , when the 
lines of curvature are parametric, satisfy the condition 
'bu 
lb_ 
dv 
D" ,= _ 
du \ E 
D 
-=-^^G V , 
\E G ^ J 
= 0, 
(5) 
where V and V are functions of u and v respectively. 
If X, Y, Z; Xi, Yi, Zi, Xi, Yi, Zi denote the direction-cosines of the 
normal to a surface C, and of the bisectors -of the angles between the 
tangents to the parametric curves, we may write the equations of a 
transformation K in the form (cf. Transactions, loc. cit.) 
1 
(a.-Xi+6,.X2 + a.,.X), 
(6) 
where mi is a constant and ai, hi, coj are functions satisfying the com- 
pletely integrable system of equations 
da,- 
bu 
bai 
dv 
bbi 
bu 
bv 
bWj 
du 
mi Chi- pBi) Ve cos 0) + hi A + 
2V-EC0SW 
nii (Xi+ pBi) VG cos CO - hiB + 
mi (X/— pOi) ^/E sin CO — ai A = 
w,B' 
2VG cos CO 
WiB 
I'VE sin CO 
// 
(7) 
mi (\4- pB^) VG sin CO + + -iB^ 
iVE sin CO 
B^/ ai 
2 V £ ^ cos CO sin 
bw,__ B" ( ai 
dv 
2VC V 
+ 
cos CO 
— )• 
sm CO/ 
where 2co is the angle between the parametric lines on C, B and B" 
are the second fundamental coefficients of C, and 
blog -y/^ 
G bv 
sin 2c 
boo 
bu 
bu 
bv ' 
