685 
According to the supposition in is a congruence of lines of curvature 
for the Vp; being Liz in the considered Vz, so that according 
to (38): 
cots Wissing vele servo le ue (UO) 
in which oz is a still unknown coefficient. Hence we conclude 
from (76): 
opl 
zin! Vir@kint 2 umij, . . . … « (77) 
j 
in which gz; are still unknown coefficients. So it is supposed that 
it must be possible to arrange i,,...,in_) in such a way that the 
equation (77) is satisfied in the same time for all values =1,...,n—1. 
Since 
iz iv = 0, kb =1,... 25m, AE) 
we find by application of in. V: 
vin A pi Thott 2 WT 8 ver bet eon ot UE) 
For £<l we have thus from (77), (78), and (79): 
icin? Vi =0, ARNE ee es ein) 
hence: 
Ur =O k=1,....,n—l (81) 
ee Zl : 
By this the equations (77) pass into: 
Kin! Vip = Okin ele er (82) 
which can geometrically be interpreted in such a way that i, is a 
congruence of lines of curvature in each of the n—1 given V,_4- 
systems. 
By application of i,.V we conclude from (78): 
Wie? Wis Win leed Mall, G3 (SS) 
Now i is V,~,-normal, hence Vi, is symmetrical in & and n, so 
that we have from (80) and (83): 
yi byes NY Se =d) ie Ma sds js ve (SE) 
hence in is V, y-normal and i,,...,i,-, are the congruences of the 
lines of curvature of the V, 4 Li. 
Since i,,...,in-, are V,_y-normal and matually perpendicular, 
we have also from (67): 
in? Vis= 0, lino op U | 5 ag oo (85) 
so that i,,...,in are the congruences of the lines of curvature 
for each of the m systems Li,,..., in. 
For a V, the proved theorem can be expressed in this way: 
When two mutually orthogonal systems of surfaces intersect along 
a congruence of curves, which are the lines of curvature of one of 
