577 
AgBsCo 18 an exscribed homologue of anc; and these two new triangles are 
also homologous to each other: the line a’’z'’c'’ being still the common 
axis, and the point p, being the common centre of homology. And the 
same thing holds good for any one of these four triangles, A,B,Co, ABC, 
a’sic’, a/’3/’'cl", in the plane II, here considered, as compared with the 
triangle ,"B,"c,", whereof the corners are those three points P,,3, which 
are not ranged on the line anc", as the three other points P,,3, namely 
a*’, Bi’, c’, have been seen to be. 
[123.] It was remarked in [28.], that each of the five pyramids z, 
is not only inscribed in the corresponding pyramid x, [26.], but is also 
homologous therewith; the centre of their homology being a point P,: thus 
the point £ is such a centre, for the two pyramids ascp and A,B,C,D,, or 
_ for those which we have lettered as © and w’ [26.][113.]. The planes 
BCD, B,C,D,, of two corresponding faces, intersect in the line c,'B,'a"; the 
planes cap, ¢,4,D, in 4’,0’,8”; the planes aBD, A,B,D, in B,A,’a”; and the 
planes aBc, 4,B,¢, in 4/’B’c/’. Hence it is easy to infer that these six 
points Po, ,, namely 
a’, BY, 0!', Aa’, Bo'y Co's 
are all situated in one plane, which is the plane of homology of the two 
pyramids © and x’, and which we shall denote by [x]; its quinary sym- 
bol being 
[e] = [11114], 
which may also serve as a type of the group [a] . - [Ez]. And in fact, 
the quinary symbols of the six points all satisfy the equation (comp. 
fio], 
ot+yt+zt+w = 4v. 
[124.] It may be noted that the two planes of homology, [p] and 
[x], have the line a/’s’’c" for their common trace on the plane anc; and 
that the traces of the three other planes. of the same group, [4], [8], [e]; 
which have 
[411], [141], [114], 
for their ternary symbols, pass respectively through the points a*, B*, C*, 
(comp. [99.]), and coincide with the lines 3,*" c,", &c., or with the 
sides of the last mentioned triangle [122.]. And it follows from [123. ], 
that the ten points P.,, are ranged sx by siz, and that the ten lines A3,, 
are ranged four by four, in five planes 15,15 namely, in the five planes 
[a]. - [2] of homology of pyramids. But these last laws of arrangement, 
of points and lines, must be considered as included in results which 
have been comparatively long known, respecting transversal* lines and 
planes in space. 
emetee sane 8. 2 
* Compare the second note to [1.]. 
BR. I. A. PROC.—VOL. VII. 41 
