692 
12. Mutually orthogonal V,—1-systems through a given congruence, 
the canonical congruences being not singly determined. 
When the roots of (24) are not all different, these roots determine 
in general q mutually perpendicular regions R,,,... Rp, Within 
the region A, every set of pz mutually perpendicular directions 
satisfies the canonical conditions. The equations (47—51) teach us 
that if must be possible to choose the canonical directions in each of 
the regions A, in such a way that they are V,_;-normal, when 
through i, there shall pass n—1 mutually orthogonal V,,—1-systems. 
Thus the conditions (C’) and (D), depending on (55) resp. (67), i.e. 
of the being WV, {-normal of al/ canonical congruences, will no more 
remain valid without any restriction. 
VG ail, 6 0 6 rg are the unit-p-vectors belonging to the regions 
I@pvein oo Ry the equations: 
ind Vye=0. (117) 
pit Vy2=0 a=l,…, B—1, BH Ung 
must be satisfied by pz independent solutions. On account of (B) 
we thus have: 
(in phere vaak vate pd? Vm pJd=0. … . « (118) 
and from this we conclude: 
psa OVA pe Oh ne oee Gero ol roer (ID) 
AART Bo. (EO) 
in which i; belongs to another region than it and ij, and for the rest 
the choice is arbitrary, provided k  /. 
(119) has entirely the same form as (55) and from (120) follows 
for the special. case that i;, iz, iv each belong to different regions: 
Ve EO WA SGD eva tte con, ol (ZI) 
an equation of the same form, and deduced in the same way as (67). 
The equations (C’) and (D) only remain valid under the above- 
mentioned restricting conditions. They are besides no longer sufficient. 
A supplementary condition will be found in the following way: 
The equation (65) shows: 
(4% — Aj) hiv? vy ij + xisin iS V *h= 0, 
(122) 
(A — 25) Wi? Vi; + xi; viz? V *h — 0. 
valid for the case that i, and i/ belong to the same region and i; 
to another one. Then, subtracting the equations (122) one from 
the other we conclude, in connection with (121): 
