603 
4 4 
6.2 27 (9 ~ in)! (V in) = 2 *h! Ph + xn un + gn? 2 Th! ‚h, (61) 
we have: 
4 
Pp Enne Vb 2 *hi sh) — 27h th (62) 
Since *h! *h has the same principal directions as *h, we may 
express the first condition also in another way, viz. that 
4 
gn 2 {(n- V) *h—2 Th! oh} 
has the same principal directions as *h: 
iin? {Gn V) *h—2 Fn! MOF EE 1, 2,..,n—-1.| (C) 
(C) ean also directly be found when we start from (30) and reason 
in the same way as we did when deriving (C’). 
In order to get a second condition we apply V to (30) and after 
that we transvect with i,. This gives: 
(Gv. V) ij} ie Ph + YG - V) ix} ij 2 7h + iin? Gd. V)*h=0 . (63) 
or, according to (29): 
Miki? Vij + 47 ij)? Vin Heijin? (iu. V)"*h=O0 . . (64) 
or: 
(AA) iris? Vi; +xijiz? (vu. V)*h=90. . . . (65) 
The vectors ij, i, and iy being all V,,-1-normal and mutually per- 
pendicular, so that 
iki? VG? Vij= gi? Vi =—iki;? Vij= 
9 
| (66) 
=a Vip Vil NE VA 
or: 
Oe LAE NOY. (67) 
the equation (65) is equivalent to 
ic? i. VI =O, 7B Allin hk =D eg nd 
(D) 
Since (Jij) | i; and in consequence (i,.V) V (ij. ij) is zero, so that: 
(MEAN Ip TET MGE 
also the principal directions of *h are singly determined. 
The tensor zdsij1 7 *h being the geodesic differential of *h, when 
moved over ds in the direction of i;, the second condition (D) expresses 
that by an infinitesimal translation in a direction perpendicular to 
i, and perpendicular to m <n—2 of the canonical directions belonging 
to ín. the component of the geodesic differential of *h im the Ry, 
determined by these m directions, has principal directions coineiding 
with m of the principal directions of *h. 
