560 
[78.] The total number of distinct lines, above obtained, is 261 + 
126 +154+124+18+383+4+3+438= 486; and the total number of their 
ternary types is 81. But 2f we set aside (as conducting to no general 
metric relations) all those lines which contain fewer than four points, there 
then remain only forty-nine lines, and only twelve types, to be discussed, 
with reference to harmonic (or anharmonic) relations, of the points upon 
those lines. 
[74.] For the purpose of studying completely all such relations, it 
will therefore be permitted to confine ourselves to the three first typical 
Lines, Bo, Aa’, B’c’, or [100], [011], [111]; the four other typical traces, 
apo’, aa”, pa”, aco, or [111], [011], [211], [211]; and fie new 
typical lines As, connecting each at least four points: namely the two 
Hines, [021] and [021], of [60.], whereof each connects the given point 
A with four points P,; and the three lines [124], [124], [112], of [64.], 
of which each connects four other points Pp, among themselves, but does 
not pass through any point Pp, or P,. 
Part [V.—Applications to the Net, continued: Harmonie and Inwo- 
lutionary Relations, of the Points situated on the Twelve Typical Lines, in 
a Plane of First Construction. 
[75.] Commencing here with the examination of the last typical 
lines, because they contain only fowr points each, let us adopt, as tem- 
porary symbols, of the iteral kind, the ten following : 
é@ =(210), 6 =(211), ¢=(021), d =(281); 
b’=(211), e =(021), d’ = (281); 
a”"=(201), 6’=(110), d"= (111); 
instead of the more systematic but less simple symbols, co” 4” 4’ 3," c™ 
AN; AtarsBy iC, iCo- 
[76.] The three lines referred to [ 64. ], are then the three following : 
abed; abled’; al!b"c'd’’. 
And because we have (comp. [16.]) the six symbolical relations, 
(e) -@=@)5 O+@M= 
(2) -(@)= (0%); @+@)= @); 
(a") — (6) =2(8");  (a")+ (¢) =2(a"), 
it results (31) that the three harmonic equations exist : 
(abed) = (ab'e'd') = (a'b"c'd") =- 1. 
We have therefore this Theorem :— 
“« Hach of the 150 lines Az, which connect four points P,, im any one 
of the ten planes {1,, and pass through no other of the 305 points Po, P,, Ps, 
2s harmonically divided.” 
