581 
[131.] For example, in the case of the I" surface (129.], this pole 
x is the point (1111) = (20201), which belongs to the group P,,,; and 
because it is on the plane [z], that plane touches the surface in that 
point: so that when the point & is the mean point of the pyramid axcp, 
the surface becomes a ruled paraboloid. In the case of the II"* surface 
[129.], the pole x of [x] is always the point (1100), or c’; this point c’ 
becomes therefore the centre of the surface, when E is the mean point of 
the pyramid; and the five following lines, 
til vii 
AB, a’B™,, B/A™, a.F, and B,G, 
where ¢ is the new point (2101) of the group P,,,, which are always 
chords through c', become in that case diameters. It may be added that, 
with the initial arrangement [91.], the surface last considered becomes 
the sphere, which is described with an for diameter ; and that it always 
passes through the auxiliary point p, of third constfuction, which was 
mentioned in [ 100. }. 
[132.] We have then here an example, of a surface of the second 
order, which was determined so as to pass [129.| through nine net-poinis 
aay a BA A Ar 
but which has been subsequently found to pass also through at least four 
other points of the net, namely 
B,, Bs, G, and Pp. 
This is, however, only avery particular case of a much more general 
Theorem, with the enunciation of which I shall conclude the present 
Paper, regretting sincerely that it has already extended to a length, so 
much exceeding the usual limits of communications designed for the 
Proceedings* of the Academy, but hoping that some at least of its pro- 
cesses and results will be thought not wholly uninteresting :— 
« Tf a Surface of the Second Order be determined by the condition of - 
passing through nine given points of a Geometrical Net in Space, it passes 
also through indefinitely many others: and every Point upon the Surface, 
* Some of the early formule of this Paper are unavoidably repeated from a communi- 
cation of the preceding Session (1859-60), but with extended significations, as connected 
now with a gwinary calculus. And ina not yet published volume, entitled ‘‘ Elements of 
Quaternions,” the subject of Nets in Space is incidentally discussed, as an illustration 
of the Method of Vectors. But it will be found that the present Paper is far from being 
a mere reprint of the Section on Nets, in the unpublished work thus referred to: many 
new theorems having been introduced, and the plan of treatment generally being different, 
although the notations have, on the whole, been retained. Besides it was thought that 
Members of the Academy might like to see the subject treated, in their Proceedings, 
without any express reference to quaternions : with which indeed the ets have not any 
necessary connexion. 
R. I. A. PROC.—VOL. VII- 4mu 
