569 
struction, containing fifteen points, one on each of the fifteen lines A, ,; 
although, in the present Paper, we can only allude to such new points 
P;, and cannot here attempt to enumerate, or even to classify them. 
[100.] We have thus again six points, at this stage, to consider, 
namely the points a, 4’, D,, 4’”, Aj, A*; and their symbols easily show 
that they are connected by the three following harmonic equations, 
(aa’D,A/”) = (4D,4Ap) = (4/40, 4%) = -— 1; 
from which it follows, by [85.], that the two triads of points, 
AA’D, and A*A’’A,, 
are triads in involution: with, of course, all the properties which have 
been proved, in recent paragraphs of this Paper, to belong generally to 
any two such triads. As a verification, it may be mentioned that, with 
the particular arrangement [91.] of the five initial points a .. 5, if we 
determine two new points P, P’, of third construction, by the formule, 
Pp (214) = Be’/"cal”’, P =|(241) >on Bal, 
it can be proved that each of the five successive intervals (comp. [ 92. ]) 
between the six points, 
wie ’ 
A, A’, Dy, iN A, Ao, 
subtends the third part of a right angle at each of these two new auxi- 
liary points, Pand p’. But with other initial configurations, the coordi- 
nates of these two new vertices would be different, because they are 
connected with angles, which are not generally projective [90.]; al- 
though, as has been already remaked, there would always be some new 
points P, or rather a circle of such, possessing the property in question. 
[101.] We may however enunciate generally, and without reference 
to any such particular arrangement of the five initial points, this 
Theorem :— 
“On any one of the fifteen lines A,,,, of second construction, and first 
group, the given point P,, and the two derived points of first construction 
P,, compose a triad, the triad in involution to which [85.] consists of the 
point P3,,, of third construction and first group, and of the two points Po, 
of second construction and second group, upon that line;’’ with seven invo- 
lutions of segments (comp. [ 84. ]) included under this general relation. 
For example, on the line aa’, the three segments aa*, 4’a’’, DA, form 
always an involution of the ordinary kind, with its double points imagi- 
nary; the three other sets of segments, AA*, A’Ag, D,A/”; A’A””’”, AAJ, D,A*; 
and p,A,, Aa’”’, a’a*, form each an involution, with real double points; the 
points a, a* are the real foci of a fifth involution, determined by the two 
pairs of segments 4’D,, and a’/’s,; the points a’, a’” are, in like man- 
ner, the real double points of that sixth involution, which the two other 
pairs, A, D,, and A,, a*, determine: and finally, p, and a, are such points, 
for the seventh involution, determined by aa’, 4’/’a*. 
R. I. A. PROC.—VOL. VII. Ak 
ny 
