( 595 ) 
in the indicated way to the point ¢, which we name the spheres 
of JOACHIMSTHAL of this point, pass through two fixed points ta, ts 
and their number is twofold infinite, it being possible to assume 
arbitrarily besides ¢, still two of the other four points t. 
2. The affinely-related point-fields (P) and (M). If the point 
P, through which the five normal planes pass, describes the normal 
plane of point tj, and ¢, as was assumed above, is always one of 
the five conormal points, the centre M of the sphere of JOACHIMSTHAL 
passing through the four other points moves in the plane that 
bisects orthogonally the distance of the points ¢g,¢; belonging to 4. 
Indicating the first plane by zr and the second plane by w, we 
have the theorem: 
“The point-fields (P) and (M) in the planes z and 4 corresponding 
“with each other are affinely related.” 
From (3) ensues 
2inmn= = t, 1-2 Yn= = t, 2 am= = t, 
6,5 6,4 6,3 
where the sums refer to six conspherical points. 
If ts, tj appear among these and if we call the others again 7, 
To, T3, Tg, WE find 
2 am = ty ts VT + (ty + ts) = 7 
4,8 4,4 
1—2 ym = ty tg ST + (to + ts) Sr + zr Site (8) 
4,2 4,3 4,4 
2am=totz Po +(tg +t) ET Er 
2 4,3 
=~ 
> 
Moreover, if ¢ is conormal with 7, Tt. 73, F4, we obtain accord. 
ing to (Ll) 
4,1 | 
2 
De 
4,1 4,2 
Er 2Er=e a Pal erste (9) 
4,2 4,3 f 
hErtEr= lg) 
4,3 44 
1 
tat ner 
4,4 3 l 
