563 
are also satisfied ; and the three pairs (or segments), 
aa’, bb’, ec’, 
which connect corresponding points, compose an tnvolution.* 
[83.] Under the same conditions, the two points a and a’ are har- 
monically conjugate to each other, not only with respect to 6 and e, but 
also with respect to b’ and ¢’; they are therefore the double points (or 
foct) of that other tnvolution which is determined by the two pairs of 
points, be, b’c’. In like manner, 4, 0’ are the double points of the invo- 
lution, determined by the two pairs, or segments, ca, c/a’; and ¢, ec’ are 
the double points of the involution determined by ad, a’b’. 
[ 84. ] From any one of the three last involutions [83. ], we could return, 
by known principles, to the involution [ 82. ]; we can also infer from them 
that the three new pairs of points (or segments of the common line), aa’, 
bc’, cb’; the three pairs, or segments, bd’, ca’, ac’; and the three others, 
cc’, ab', ba’, form three other involutions, making seven distinct involutions 
of the six points, so far : in three of which, as we have seen in [ 88. ] two 
of those six points are their own conjugates. 
[85.] For these and other reasons we propose to say, that when any 
three collinear points (as a, 6, ¢) are assumed (or given), and three other 
points on the same line are derived from them, by the condition that each 
shall be the harmonic conjugate of one, with respect to the other two, then 
these two sets of points are two Triads of Points in Twolution. And it 
is easy to extend this definition so as to include cases of two ¢riads of 
complanar and co-initial /ines, or of collinear planes, which shall be, in 
the same general but (as it is supposed) new sense, in involution with 
each other: every such involution of triads including, by what precedes, 
a system of seven involutions of the old or usual kind. 
[86.] For example, because the two triads of points, a’B/c” and 
A” Bc”, are thus in involution, by the equations [ 81. ] applied to the 
fourth typical trace [48.], it follows that the two pencils, each of three 
rays : 
; D, . Apc’, and D, . ABC, 
are triads of lines, in involution with each other ; and that, for a similar 
reason, the ¢wo triads of planes, all passing through the line pz, 
DEA, DEB, DEC, and DEA”, DEB’, DEC”, 
are, in the sense above explained, in ¢nvolution. In fact, when the point 
D, is thus taken as @ vertex of the pencils in the plane axzc, the three har- 
monic equations of the first case [ 81.], namely, 
{c"a''B’A™) = (aleieE*) = (B’c"a"c") ae 1, 
4 Compare p. 127 of the Géomeétrie Supérieure (Paris, 1852). In general, the reader 
is supposed to be acquainted with the chapter (chap. ix.) of that excellent work of M. 
Chasles, which treats of Involution. 
