1422 
complex reflections in one or more planes). This is a consequence 
of the linear-orthogonal character of the substitution by which only 
congruent and symmetrical figures are possible. 
For the same reason we must also assume that each imaginable 
space-time-A of the considered kind may be considered as a complex 
reflection in one or more symmetry-spaces. This shows at once the 
way in which each space-time-symmetry-element can be found. 
§ 5. General formula for the coordinates of the point B, found 
from the point A by application of an arbitrary complex space-time- 
symmetry-operation. of the above considered kind. 
k=m 
on =d 2 Bea! UPE EPR U Ok DoY Bo) [PE + 
l=m 
+ ERG (Q, k) + = (1, p)(p, 4) + & (Lp) (p, 9) | 4) +) Viart) 
where (p,q) has been written for 
— Zp! Pq + Pp Pa? + Pp Pq? + Fp* Pa‘) 
while we must take 1 >p >q > ete. 
§ 6. We can derive all space-time-symmetry-operations by simply 
combining, all space-symmetry-operations that have been mentioned 
without the limiting to 2-, 3-, 4- or 6-al axes with time-symmetry- 
operations. The linear orthogonal substitution, expressed by formula 
(5) will have a scheme of coefficients : 
1% 145 1%3 1% 1 
2%) 2% 27 24 ot 
si zl, 323 314 3% 
Gh Gig gla? 4G, Gas 
while these coefficients are connected by the following relations: 
14,7+ ,4,7+,4,°+,a7%=1) 14, 14, +44, 243 +,4, ,4,+,4, 4,0: 
a + ,a,7+ ,4,°-+ 0,7 =1 pi ay 1%, 143 12% 9% 13% 9%, 1,4, 2, — 9 
A Hea Hide +44 =l | 12, 144 Ha 24, +,4, ,a,+,0,,¢,—09 | 
a Hea Hira Ha =l VAE +48, 7%, + 40, A + 2,40, =0 
1% 14e Hr aq al ty Fe staat 1% = LO 
1%; ,4,+,4, a, +,4, ,4,+,2, ,a, =O 
1) (Note added during translation). See f.i. G. Viota N. Jahrb. f. Miner., Geol. 
und Pal. Beil. Bd. 10 p. 495 1896. G. Wurrr Zs f. Kryst. u. Miner. 27 p. 556 
1896. 
