548 
which also they are homologous ; the corresponding centres of homology 
being the ten points P,, in the same order, 
A’, &.; Ao, &e.; A,, &c.; and Dj, as before. 
The axes of homology of these ten pairs of triangles 1,, T,, will offer them- 
selves a little later, in connexion with points P,. 
[30.] All this may be considered as evident from geometry alone, at 
least with the assistance of literal symbols, such as those used above. 
But to deduce the same things by calculation, with quinary symbols and 
types, on the plan of the present Paper, we may observe that the sym- 
bolical equation, 
(10000) + (01100) + (00011) = (11111), 
considered as a type of all equations of the same form, proves by (18) 
or (72) that each point P, can, in three different ways, be combined with 
another point P,, so that their joining line shall pass through a point Py; 
and that thus the group of the 15 lines A,,, arises, of which the line 
Aa’D, is a specimen, and may be called a Second Typical Line (the first 
such line haying been zc, by [28.]). The complete quinary symbol of a 
point on this line is (twwvv), which is however congruent to one of the 
form (¢uw00), and may therefore be abridged to the ternary symbol (tuu), 
or (yy); and the quinary symbol of a plane through the same line is of 
the form [Ommrr ], or [Ottuu]; we may therefore, by [21.] (comp. [28.]) 
consider the two expressions, 
(cyy), and [0ttuz], 
as being not only local and tangential symbols for the particular (or typi- 
cal) line aa'D, ttself, but also local and tangential types for the group Ao, 13 
or as the pornt-type, and the plane-type, of that group. 
[31.] The two points P,, of which the quinary symbols have been 
thus combined in [30.], had xo common coordinate different from zero ; 
but there remains to be considered the case, in which two points of that 
group have such a coordinate: for example, when the points have for 
their symbols, 
(10100) and (11000), or (101) and (110). 
The potnt-symbol and plane-symbol of the line A, connecting these 
two points P, are easily seen to be (with the same significations of c ands 
as before), 
(otu00), or (ctu), and [tttue]; 
but no choice of the arbitrary ratio, ¢: u, with o=¢+4u, will reduce the 
symbol (tw) to denote any one of the 15 points Pp, P,, except the two points 
P, (in this example, B’ and c'), by joining which the line is obtained ; 
considering therefore the two last symbols as types, we see that they re- 
present a second group, consisting of thirty lines A,,,; but that there 
can be no third group, of lines A, of second construction. The particular 
line ’c’, which the symbols in the present paragraph represent, may be 
