687 
and this is exactly the equation given for the first time for R, by 
LitientaaL*), and to which lately, also for R,, Wierinea *) has again 
drawn the attention. So this condition is a special case from Ricct’s 
first. It remains also valid for an arbitrary linear element, and 
also for n > 3, then however it is no longer the only condition. 
9. Rreer’s conditions. Be i, again V,_,-normal. Then we can choose 
an original variable y” and vectors s„ and s’„, so that:' 
: 1 , 
In = On Sn = — Sn Ot Oe vei On BPL (89) 
On 
By means of this equation we can eliminate in from (C) and (D) 
and substitute s, for it. 
Since: 
(in : V) (V =A in) => (in : V) {On V sn + 3 (V On) Sn + 4 Sn Von}, . (90) 
we have: 
4 4 
gn? (in. V) (V — in) = gn 2 { (in -V On) V sn + On ini VV sn + (91) 
+ 4(V on) int V su + b (in? V sn) V on}, 
or, since: 
WV on == V (sn. Sn) % = — XOn° (VSn)* Sn —=— On Un -+ X%On Sn! (Von) sn, (92) 
also: 
4 4 
En? (in. VY) (VY — in) = En? § (in. Von) V Sn + On? sn! VV Sn—xttn Un}. (93) 
Since on account of (31) and (69): 
bir? {2x7 (TV ~ in)! Vin =ijik? unum, . . . (94) 
the condition (C') gets the shape: 
iig }%On? Sp} VV Sn — 2 un that == 0) Saws (C,) 
Since: 
1 1 
Va=—Vh+(V—) In, artes st) Listes (95) 
we further have, in connection with (30) and (33): 
ij ir 2 Vsn = 0, . . Oe ey ur oF i e (96) 
from which by application of (ix.V) may be concluded: 
] 1 3 
— (i; 1 Vig) 4 (V in) Lig + — (itt V ix) t (V in) * ij Hij ici: ? VV sn—0. (97) 
Gn In 
h R. vy. LittentHar, Ueber die Bedingung, unter der eine Flaichenschar einem 
dreifach orthogonalen Wlächensystem angehört. Math. Annalen 44 (94), 449—457. 
4) W.G. L. Wierinaa, Over drievoudig orthogonale oppervlakkensystemen. Diss. 
Groningen, (18) 59 pp., see p. 13. 
5) See note bj of next page. 
