689 
euclidean space the condition (D',) has been given by Dargoux ') *). The 
4 ~ 
characteristic numbers (lljn) of K vanish too, when the Vs Li, 
are geodesic. 
10. Lé&vy’s, Caytry’s and DarBoux’s conditions. Differentiating the 
relation: 
Deonet AO ie Ul Rha CTOS) 
we get 
V7 Ie => (V Gn) Sn + On WS nee sie 2 ors (104) 
Differentiating again, we get: 
VV in = (VV Gn) Sn + (V Sn) 1 a(V Gn) a + (V on) V sn + On VV Sn, (105) 
and. from this and (104) we have for VV on: 
2x 
“VV on = On (VV in) 1 in — Gn? (VV sn)! in + (V on) (WV on). (106) 
Since: 
(VV in)! in=V {(V in) 1 in} — (V in) 1 a (Vin) 1 a=~*h 1 *h-(un. un) in in,(107) 
we get, in connection with (92): 
4 4 
Én 2 VV On —= — On ‘ht *h— x0n’ En 2 (VV Sn) 1 Sn ++ 2 On Un Un. (108) 
In connection with (C,) this equation gives a new shape to the 
first condition : 
1) G. Darpoux, Legons sur les systèmes orthogonaux et les coördonnées curvi- 
lignes I (98), p. 130, form. (35). 
2) As a simple example for the application of (C,) and (D‘;) for euclidean space, 
we can take the system u = Y, (y!)-++...-+ Yn (y"), in which y!,..., yn are Cartesian 
coordinates. To calculate gaz etc. it is necessary to find a system of m—1 Vn—1 
which determines in the V;,—1 U = const. a system of coordinates ea,... Then 
Kea, . Ca, — Yaya etc. For this purpose we must try to find »—1 independent 
solutions of the differential equations : 
TRAAN eco apn! 
Se ey aaa Sh Gi) le 
5 Oy? dy 5 RO 
For the calculation compare e.g. WiIeRINGA, Diss. p. 21 and seq. Then we can 
see that the condition (D’)) is identically satisfied, so that only LartenrnaL’s con- 
dition (Cj) remains, which can be written in this case: 
1 ] 1 
iy Vark verd 0, 
Y/Y /" —2 Yi PY —2 ¥/'? Ye Ye — 2 A 
or Yi Yi"—2 Yi’? = AYi’ + B, in which A and B are constants. 
This result has been deduced for n —=3 by Street, and for a general n by 
DAgBOUX in another way as has been done here. Comp. Darsoux, Legons sur les 
systémes orthogonaux etc., p. 140 and 141. 
