1421 
& point’ A(c,', 7,7, 2", 2,‘) into a point “B(a,', 2,7, 7,", v,‘) that is 
geometrically to be found in the following way. 
Draw through A a perpendicular to t and measure the length of 
that perpendicular on the other side of its point of intersection 
with r. The end-point of this stretch is the point 5. 
When X is given by the length /, of the perpendicular from the 
origin of R, on t and its direction cosines ~,', p,°, p‚° and p‚*, where 
Pt PET Oy eee oy! Sy 
then we find by substitution of one of the four indices 1, 2,3 and 4 
for n 
tr == + 2 Pi" (,—9,' DP BP, EP, x, ") GP ee (2) 
When in a three-dimensional w'2’2?-system we consider a plane 
V through the origin, the direction cosines of its normal being 
in the ratio p‚*, m,? and g,*, then the points A’ and 5’, corres- 
ponding in this system to A and B in the four-dimensional system, 
are lying on the same perpendicular to V, «,»—.,” being proportional 
to gy," for the values n = 1, 2,3. 
When the distances from A’ and B’ to V are denoted by —y» 
with m==1 and 2 resp., then we have 
9," an A y,* Em + y,° Bin” 
Vlet 
Im = (3) 
and therefore 
Ys Fad waa: 2 V1—g? (;-—- V1—o?? YP a,*) (4) 
a," = a," + 2 Pr" Lt Me Y:—Pr a, *) 
Thus the distance from B’ to V and the new value of ict are 
evidently found from the values of these quantities for the point- 
incident A by drawing in a two-dimensional ya*-system a line in 
such a way, that the perpendicular to it from the origin has a 
length 7, and forms with the z2* axis an angle with a cosine= 9,‘ 
and by reflecting the point A" with the coordinates x,‘ and y, in that 
line. The coordinates of the point B" thus found give the new value 
of the time and the new distance to V in the three-dimensional figure. 
§ 4. Hach space-time-symmetry-operation of the constdered kind 
may be regarded as a complex symmetry-operation of space-time- 
reflections. 
The A’s treated by ScHoENFLIEsS and predecessors, reflection in 
a plane, inversion about a centre, translation, rotation through 
Ln tag a and thd 2-, 3-, 4- and 6-al screws, 2-, 3-, 4- and 6-al 
ee a ee 6 
rotatory-reflections and gliding reflection may all be regarded as 
