575 
{117.] As a specimen of a case of collineation which conducts to 
such new relations, let us take the four following points P,, in the se- 
cond typical plane, 
a = (01120), b = (00211), ¢ = (02031), d = (01302), 
whereof the two first are points P,,,, and the two last are points P,, 5; 
and of which the symbols satisfy the equations, 
(c) = 2(a) — (b), (d) = — (a) + 2(6); whence (adbc) = 4. 
These four points, therefore, with which it is found that no other given or 
derived point of the system P,, P,, P. is collinear, do not form a harmonic 
group; and consequently we cannot construct the fourth point, d, when 
the three other points, a, b, c are given, by means of harmonic relations 
alone (comp. [108.]), unless we introduce some aua«iliary point, or 
points, e, . ., which shall be at lowest of the third construction. But if 
we write 
e = (12020) = (01111), f = (10220) = (01381), 
so that e is a point Ps,, [99.], while f may be said to be a point Py,., 
we find that these two new or auxiliary points, e, f, are the double points 
of the involution, determined by the two pairs, ab, cd; because we have 
the two harmonic equations, 
(aebf) = (cedf) =- 1. 
And because we have also, 
(cabe) = (abde) = - 1, 
we need only employ the one auxiliary point e, considered as the har- 
monic conjugate of a, with respect to 6 and ¢c; and then determine 
the fourth point d, as the harmonic conjugate of a, with respect to 6 
and ¢. It may be added that abe and def are triads in involution [85.]; 
so that if e be projected to infinity, the finite line cd is trisected at a 
and 6. 
Part VI.—On some other Relations of Complanarity, Collinearity, Con- 
currence, or Homology, for Geometrical Nets in Space. 
[118.] Although we have not proposed, in the present Paper, to 
enumerate, or even to classify, any points, lines,' or planes, beyond what 
we have called the Second Construction [1.], yet some such points, lines, 
and planes have offered themselves naturally to our consideration: and 
we intend, in this Sizth Part, to consider a few others, chiefly in con- 
nexion with relations of homology, of triangles or pyramids which have 
been already mentioned. 
{119.] It was remarked in [29.], that the thirty lines A,,, are the 
sides of ten triangles 1,, of second construction, which are certain inscribed 
