562 
belong to the same two types as the two last, would offer to ovir notice 
a pencil of four rays, which has some interesting properties, especially 
as regards its intersections with other pencils, but which we cannot here 
delay to describe. 
[80.] It may, however, be worth while to state here, as a conse- 
quence from the preceding discussion, this other Theorem :— 
“‘ The 120 lines Az, in the ten planes II,, whereof each connects a point Py 
with four points Ps, and with no other of the 305 points, although not all 
syntypical, are all homographically divided.” 
[81.] Proceeding to consider the arrangements of those six typical 
lines [ 58. | which contain each six points, we find that whether we write, 
as new temporary and literal symbols, 
a= (011), 6=(101), ¢=(110), a = (211), = (121), ¢ = (112), 
or @= (011), 6 = (111), ¢ = (120), a’ = (281), B! = (181), e = (102), 
the six points abca' b/c’ being in the one case on the line [111], and in 
the other case on the line [211], we have in each case the three harmonic 
equations : 
(caba’) = (abeb’) = (beac) =-1. © 
We may then at once infer this Theorem : 
“‘ The 70 lines Az, in the ten planes T1,, which are represented by the 
fourth and seventh typical traces of planes on the plane axzc, although not 
all syntypical (or generated by similar process of construction), are all 
homographically divided.” 
[82.] This common mode of their division may deserve, however, a 
somewhat closer examination, its consequences being not without inte- 
rest. When any six collinear points, a .. ¢’, are connected by the three 
equations [81.], we are permitted to suppose that their symbols are so 
prepared (if necessary), by coefficients,* as to give, 
(2) +(@)+(@=9;- 
(2) =(6)- ©, @)=@-@, @)=@ - ©) 
and therefore, 
(a) + (0’) + (¢’) = 0, 
3(a) = (¢) - (8), 8(8) = (#) - (“), 8) = (0) - (@). 
Whenever, then, the three harmonic equations [81.] exist, for a system 
of si collinear points, a. . c', the three other harmonie equations, formed 
by interchanging accented and unaccented letters, 
(e'a’b'a) = (a/b'e'b) = (b’e'a’c) = - 1, 
* For example, in the second case [81.], we should change the symbols for c and 0’ 
to their negatives, before employing the formule of [82.]. 
