576 
homologues of ten other triangles 1,, of first construction [26.|; the ten 
corresponding centres of homology being the ten points P,. For ex- 
ample, the triangle a’s’c’ is inscribed in axe, and is homologous thereto, 
the point p, being their centre of homology ; because we have the three 
relations of zntersection, 
A’ = D,A'BC, &e. ; 
or because, a’ being a point on Bc, &c., the three joining lines aa’, &c., 
concur in the point 2. 
[120.] Proceeding to determine the aais of this homology, or the 
right line which is the locus of the points of intersection of correspond- 
ing sides, we easily see that it is the line a/’3’’c”; because we had 
a’ =xsezc’, &c. And because an analogous result must take place in 
each of the ten planes II,, we see that the ten points P,,, are ranged, three 
by three, on ten lines Az,,, in the ten planes II,; namely on the axes of ho- 
mology of the ten pairs of triangles, 1,, T., in those ten planes: which 
axes are the lines, 
D,'A,/A,', &e.; C,'B,'A", &e.; c,/B,/a”’, &e.; and apc’; 
each point P,,, being thus common to three of them, because it is com- 
mon to those three planes I1,, which contain the line A, whereupon it is 
situated. Each point p,,, is also the common intersection of this last 
line with three lines A.,2; we have for example, the formule of con- 
currence, 
A” = BC'B‘C'"B,C,"B,Cy. 
[121.] The line a’3’c’’ was seen to be the common trace of two 
planes II,,, namely of a,B,c, and a,B,C,, on the plane I1,, namely azc, in 
which it is situated; and a similar result must evidently hold good for 
each of the ten lines A;,, But we may add that the three triangles axc, 
A,B,C, AgB,C2, in the plane of each of which the line a/’s"c’’ is contained, 
are homologous, two by two, and have this line for the common axis of 
homology of each of their three pairs; having however three distinct cen- 
tres of homology, namely p,’ for second and third, p for third and first, 
and £ for first and second: with (as we need not again repeat) analo- 
gous results for the other lines A;,,, of which group we here take the 
line a’3'c" as typical. It may be remarked that the four centres, re- 
cently determined, are collinear, and compose an harmonic group ; and 
that the inscribed triangle a'p‘c' is also homologous with each of the two 
triangles A,B,C,, A,B,C,, although not complanar with either; the line 
Asc" being still the common axis of homology ; while the two centres, 
of these two last homologies, are the two given points, p and E. 
[122.] The sta points p,,,, in the plane azc, have been seen to range 
themselves, according to their two ternary types [41.], into two sets of 
three, which are the corners of two new triangles; one of these, namely 
a’’3!"c'", being an inscribed homologue of a's’c' ; while the other, namely 
