549 
taken as typical of this second group; and may be called (comp. [ 23. ] 
and [30.]) a Third Typical Line of the System, or Wet, determined by 
the five: given points 4... And the pyramids B,, Bs, and triangles 
T,, T:, of first and second constructions, of which the /iteral symbols have 
been assigned in [ 26. } [28.] [29.], might also have easily been suggested 
and studied, by guinary symbols and types alone. 
[82.] As regards the Planes 11, of Second Construction [1.], it is ea- 
sily seen that no such plane contains any ¢wo points Po, or any one line 
A,; for example, the first typical line zc [ 23. | contains the point a’; and 
if we connect it with any one of the four points a, B’, c’, D,, we only geta 
plane I1,, namely asc; if with D, 4,,B,, or c,, we get another plane I, 
namely scp ; and if with any one of the four remaining points £, A., By, C1, 
the plane BCE is obtained. Accordingly, the general symbol [200¢w ], in 
[23.], for a plane through the line zc, gives o = 0, or ¢= 0, or w= 0, 
when we seek to particularize it, by the first, the second, or the third of 
these three sets of conditions respectively. 
[33.] But if we take the symbol [O¢¢u7], in [80.], for a plane 
through the second typical line aa’D,, and seek to particularize this 
symbol by the condition of passing through some one of the eight points 
P, which are not situated upon it, we are conducted to the ‘following 
results. The points x’, c’ give ¢= 0, and the points 4,, A, give w=0; 
these points therefore give only two planes 11,, namely the two planes 
asc and apg, of which the line A,,, is the intersection. But the points 
By, C2 give ¢=u, and the points ¢,, B, give ¢=— wu; these points there- 
fore give two planes of a new group, Il,,,, namely (comp. [ 20. ]) the two 
following : 
plane aa’D,B,c, = [01111]; plane aa’p,c,B, = (01111); 
which are of the same ¢ype as the plane (96), namely, 
plane 48B,c,C,B, = [01111]. 
There are fifteen such planes II,,,, as the type sufficiently shows; each 
passes through one point P,, and contains two lines A.,,, containing also 
Sour lines A,,2; as, for instance, the last-mentioned plane 48,c,c,B,, which 
we shall call (comp. [ 24.]) the Second Typical Plane, contains the two 
lines AB,C., Ac,B, [28.]|, and the fowr lines 8,0, C,C2, C,By, B,B,; that is to 
say, the two diagonals and the four sides of the guadrilateral B,c,c,B,, of 
which the plane U,,, passes through a. 
[34.] We have now exhausted all the planes 0, which contain any 
point Pp; but there exists a second group of planes, M12, , each of which 
is determined as connecting three points P,, although passing through no 
point p. Thus if we take the third typical line x’c! [31.], and the 
symbol [¢¢twa] for a plane through it, we get indeed ¢=0, oraplane II,, 
namely, azc, if we oblige the plane through 8’c’ to contain A, OF B, Or 
c, or a’, or D,; and we get u = 0, or [11101], or a plane II,,,, namely 
DB'B,C’ , as in (95), if we oblige “it to contain D, or B,, or ¢,; while we 
