565 
pairs of segments, be, b’c’; ca, ca’; ab, ab’, are likewise real, and have 
been assigned [83.]; namely, in each of these three last cases, the two 
remaining points of the system. 
[89.] Now, in general, when the foc? (or double points) of an invo- 
lution of collinear segments, aa’, bb’, . . . are imaginary, so that conju- 
gate points, a, a’, or 6, U', &e., fall at opposite sides of the central point 
0, it is‘ known, and may indeed be considered as evident, that if an or- 
dinate ov be erected, equal to the constant geometrical mean between the 
two distances oa, oa’, or ob, ob’, &c., then, all the segments aa’, bb’, &e., 
subtend right angles, at the extremity P of this ordinate. It follows, then, 
by what has been proved in [82.] and [88.], and by the first case of 
[81.], that each of the three segments a” a, B’ BY, cc", of the fourth 
typical trace [48.|, subtends a right angle at some one point, P, im the 
plane asc, or rather generally at each of two such points: and in like 
manner, by the second case [81.], that each of the three other segments, 
A’ A™, Cy By”, C1" BY, of the seventh typical trace, subtends a right angle, 
at each of two other points, P, P’, in the same plane. 
[90.] These results, by their nature, like all the foregoing results of 
the present Paper, are quite independent of the assumed arrangement of 
the five given (or initial) points of space a . . B, and are unaffected by pro- 
jection, or perspective. In saying this, it is not meant, of course, that 
one right angle will generally be projected into another ; or that the new 
point Pp, at which the three new segments a”a™, B'B™, o’c™, or A'A™, CoBy”, 
c,"B", subtend right angles, will be itself (what may be called) the pro- 
jection of the old point vp [89.], which was so related to the three old 
segments, denoted by the same literal symbols, when the arrangement (or 
configuration) of the five ential points is varied, by a process analogous 
to projection. We only assert that there will always, in every state of 
the Figure, or of the Wet, be some point P, possessing the above-men- 
tioned property: or rather that there will be a czrcle of such points in 
space, having for its axis the line to which the three segments belong. 
[91.] To fix a little more definitely the conceptions, let a, 8, c, p be 
supposed, for a moment, to be the corners of a regular pyramid, with & 
for its mean potnt, or centre of gravity. With this arrangement of the 
five given points 2, six of the derwed points P,, namely a’, B’, C’, Ao, Ba, 
C2, bisect the sia edges, BC, CA, AB, DA, DB, DC, of the given pyramid; and 
the four other points P,, namely A,, B,, C, D,, are the mean points of the 
four faces, opposite to a, B, c, D. Six of the ten points P,,,, namely 
A”, BY, cl, Ay!, Bo’, Cx’, are now tnjinitely distant ; and the line a/c" 
A™p"c to which three of the lately mentioned segments belong, becomes 
the line at infinity in the plane anc: which might seem, at first sight, 
to render difficult, with respect at least to them, the verification of a 
recent theorem [89.]. That theorem is, however, verified in a very 
simple manner, by observing that, with the arrangement here conceived, 
the three angles sD", B’’D,B8", c/D,0", which those mfinite and infinitely 
distant segments may be imagined to subtend at the point D,, are all right 
