( 207 ) 
It remains to be proved that the relation between the five lines, 
obtained by adding this new line a; to the four given lines a is 
mutual in such a sense, that all quadruples of which a; is a line 
lead back in the indicated way to the fifth line. 
We prove this for the quadruple a, a,a,a;, and for this complete 
this to the double four 
and then verify that the four spaces (a,c), (ag c2), (a3 ¢s), (a5 ba) 
have the line a, in common. 
The lines cj, cg, cg as the lines of intersection of the triplets 
of spaces 
(ag as). 2.2. — # + % — % +H + 2 = 0 
(ay G;)hs os. Tj — Tg — a + v + a, = 0 oe « « Chy 
EN ee Ly 0 
(az a5)...» — AH & — #3 + % + % = 0 
(ay az). 1s 2 — dr Ag + ty — ty + % = Of... Gy 
(a, da) * V4, 
(a, ds) s+. — % + % + 23 — % +e = 0 
(ag Ap). 2 % — % — t+ a + a, = 0 ee « « Cg 
(a, ag)... . — er — & + #3 + um + a = 0 
are then represented by the equations 
Cy eee ee dg ty = te, % =O, my = 0 
Grp eys le 18 ty =. hg SS te, fy Se, tee 
